Formula Library

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

The straight-line distance between two points on a coordinate plane.

d

— the distance between the two points

x₁, y₁

— the coordinates of the first point

x₂, y₂

— the coordinates of the second point

When to use it: Whenever you know the coordinates of two points and need the straight-line distance between them.

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

The point exactly halfway between two given points.

M

— the midpoint

x₁, y₁

— the coordinates of the first point

x₂, y₂

— the coordinates of the second point

When to use it: Whenever you need the exact centre point of a segment joining two known points.

m = (y₂ − y₁) / (x₂ − x₁)

How steep a line is — the change in y for each unit change in x.

m

— the slope (gradient) of the line

x₁, y₁

— the coordinates of the first point on the line

x₂, y₂

— the coordinates of the second point on the line

When to use it: Whenever you need to measure or compare how steep a line is, or check whether two lines are parallel or perpendicular.

y = mx + c

Describes every point on a non-vertical straight line, using its slope and y-intercept.

m

— the slope of the line

c

— the y-intercept — where the line crosses the y-axis

x, y

— the coordinates of any point on the line

When to use it: Whenever you need to write down, graph, or predict values along an entire straight line, not just one point on it.

a⁰ = 1 and a⁻ⁿ = 1/aⁿ (a ≠ 0)

Any nonzero base raised to the power 0 equals 1; a negative exponent means "take the reciprocal, then raise to the positive exponent".

a

— the base (any nonzero real number)

n

— a positive integer exponent

When to use it: Whenever an exponent is 0 or negative and you need to rewrite it as a positive-exponent expression (or a plain number).

aᵐ · aⁿ = aᵐ⁺ⁿ and aᵐ / aⁿ = aᵐ⁻ⁿ (a ≠ 0)

Multiplying powers with the same base adds the exponents; dividing powers with the same base subtracts the exponents.

a

— the common base (any nonzero real number)

m, n

— the two exponents

When to use it: Whenever you multiply or divide two powers that share the exact same base.

(aᵐ)ⁿ = aᵐⁿ and (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ

Raising a power to another power multiplies the exponents; raising a product or quotient to a power distributes that power to every factor.

a, b

— the bases being raised to a power

m, n

— the exponents

When to use it: Whenever an expression already raised to a power is itself raised to another power, or when a whole product/quotient is raised to a power.

a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)

A fractional exponent packages a root and a power together: the denominator of the fraction is the root (index), and the numerator is the power.

a

— the base (a ≥ 0 whenever n is even)

m, n

— integers — n is the root index (n > 0), m is the power

When to use it: Whenever you need to move between radical notation and exponent notation, e.g. to apply the exponent laws to an expression that starts as a root.

ⁿ√a · ⁿ√b = ⁿ√(ab)

The product of two radicals with the same index equals the radical of the product of their radicands.

a, b

— the radicands (a, b ≥ 0 when n is even)

n

— the common index (root) of both radicals

When to use it: Whenever you simplify a radical by factoring out a perfect nth-power factor, or multiply two radicals with the same index.

ⁿ√a / ⁿ√b = ⁿ√(a/b) (b ≠ 0)

The quotient of two radicals with the same index equals the radical of the quotient of their radicands.

a, b

— the radicands (a ≥ 0, b > 0 when n is even)

n

— the common index (root) of both radicals

When to use it: Whenever you divide two radicals with the same index, including rationalizing a denominator that contains a radical.

If aˣ = aʸ and a > 0, a ≠ 1, then x = y.

Once both sides of an exponential equation are written with the exact same base, the exponents themselves must be equal.

a

— the common base (a > 0 and a ≠ 1)

x, y

— the exponents being compared

When to use it: Whenever you can rewrite both sides of an exponential equation using the same base, turning the whole equation into a simple equation of exponents.

A = A₀(1 + r)ᵗ (growth, r > 0) A = A₀(1 − r)ᵗ (decay, 0 < r < 1)

A quantity that changes by the same percentage every fixed time period follows this pattern — starting amount times a growth/decay factor raised to the number of periods elapsed.

A

— the amount after t time periods

A₀

— the initial amount, at t = 0

r

— the growth rate (or decay rate), as a decimal

t

— the number of time periods elapsed

When to use it: Whenever a quantity grows or shrinks by a fixed percentage each period — population, compound interest, or radioactive-style decay.

a × 10ⁿ, where 1 ≤ a < 10 and n is an integer

A compact way to write very large or very small numbers as a decimal between 1 and 10, multiplied by a power of 10.

a

— the coefficient — a decimal number with 1 ≤ a < 10

n

— the exponent (power of 10) — positive for large numbers, negative for small numbers

When to use it: Whenever a number has many digits or many leading/trailing zeros, and you need a compact, easy-to-compare form.

log_b(x) = y ⇔ bʸ = x (b > 0, b ≠ 1, x > 0)

A logarithm answers "to what power must the base b be raised to get x?" — it is the exponent y in the equivalent exponential equation bʸ = x.

b

— the base of the logarithm (b > 0, b ≠ 1)

x

— the argument — the number you are taking the logarithm of (x > 0)

y

— the logarithm itself — the exponent that produces x

When to use it: Whenever you need to find the exponent that produces a known result, or to switch between logarithmic and exponential form.

log_b(MN) = log_b M + log_b N

The logarithm of a product equals the sum of the logarithms of its factors.

b

— the base of the logarithm (b > 0, b ≠ 1)

M, N

— positive real numbers being multiplied

When to use it: Whenever you need to expand the logarithm of a product into a sum, or condense a sum of logarithms into one logarithm of a product.

log_b(M/N) = log_b M − log_b N

The logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator.

b

— the base of the logarithm (b > 0, b ≠ 1)

M, N

— positive real numbers, with M divided by N

When to use it: Whenever you need to expand the logarithm of a quotient into a difference, or condense a difference of logarithms into one logarithm of a quotient.

log_b(Mᵖ) = p · log_b M

The logarithm of a number raised to a power equals the power times the logarithm of the number.

b

— the base of the logarithm (b > 0, b ≠ 1)

M

— a positive real number

p

— the exponent applied to M

When to use it: Whenever a logarithm's argument is itself a power, letting you bring the exponent down as a multiplier — essential for solving equations where the variable is in an exponent.

log_b M = log_a M / log_a b

Converts a logarithm in one base (b) into a ratio of logarithms in a different, more convenient base (a) — usually base 10 or base e, so a calculator can evaluate it.

b

— the original base of the logarithm

a

— the new base being switched to (commonly 10 or e)

M

— the argument of the logarithm

When to use it: Whenever you need to evaluate a logarithm in a base your calculator does not have a direct button for (any base other than 10 or e).

ln x = log_e x, where e ≈ 2.71828...

The natural logarithm is simply a logarithm with base e, a special irrational constant that arises naturally in continuous growth and decay processes.

x

— the argument of the natural logarithm (x > 0)

e

— Euler's number, an irrational constant approximately equal to 2.71828

When to use it: Whenever a process involves continuous (rather than step-by-step) growth or decay, such as continuous compound interest or radioactive decay.

n(A × B) = n(A) × n(B)

The number of ordered pairs in the product set A × B equals the number of elements in A times the number of elements in B.

n(A × B)

— the number of ordered pairs in the product set A × B

n(A)

— the number of elements in set A

n(B)

— the number of elements in set B

When to use it: Whenever you need to know how many ordered pairs a product set contains, without listing every pair by hand.

Vertex = (−b/2a, f(−b/2a)), for y = ax² + bx + c

Gives the exact coordinates of the turning point (vertex) of a parabola directly from its coefficients, without needing to complete the square.

a, b, c

— the coefficients of y = ax² + bx + c (a ≠ 0)

−b/2a

— the x-coordinate of the vertex, and the equation of the axis of symmetry

f(−b/2a)

— the y-coordinate of the vertex — the minimum or maximum value of the function

When to use it: Whenever you need the highest or lowest point of a parabola, or its axis of symmetry, directly from the equation y = ax² + bx + c.

D = b² − 4ac

A single number, computed from a quadratic's coefficients, that reveals how many real roots (x-intercepts) the quadratic has — without solving the equation.

D

— the discriminant

a, b, c

— the coefficients of ax² + bx + c = 0 (a ≠ 0)

When to use it: Whenever you need to know how many real solutions a quadratic equation has (two, one, or none) before — or instead of — actually solving it.

x = (−b ± √(b² − 4ac)) / 2a

Solves any quadratic equation ax² + bx + c = 0 directly from its coefficients, without needing to factor.

x

— the solution(s) — the roots of the equation

a, b, c

— the coefficients of ax² + bx + c = 0 (a ≠ 0)

±

— produces two solutions — one using +, one using −, unless the discriminant is 0

When to use it: Whenever a quadratic equation cannot be factored easily (or at all), or you want a reliable method that always works.

y = a|x − h| + k, vertex = (h, k)

Every absolute value function graph is a V-shape (or upside-down V) whose sharp corner — the vertex — sits at exactly (h, k), directly readable from the equation.

a

— controls the direction (sign) and narrowness (size) of the V, same role as in a parabola

h

— the horizontal shift — the x-coordinate of the vertex

k

— the vertical shift — the y-coordinate of the vertex

When to use it: Whenever you need to find the vertex, sketch the graph, or identify the transformations of an absolute value function directly from its equation.

If q ≥ 0: x − p = q or x − p = −q, i.e. x = p + q or x = p − q. If q < 0: no solution.

Since |x − p| measures a distance (from x to p) and distance is never negative, this equation asks "which x-values are exactly q units away from p?" — there are exactly two, one on each side of p, unless q is negative.

x

— the unknown being solved for

p

— the reference point the distance is measured from

q

— the required distance from p (must be ≥ 0 for a solution to exist)

When to use it: Whenever you need to solve an equation with a single absolute value expression set equal to a number.

|x − p| < q ⇔ p − q < x < p + q |x − p| > q ⇔ x < p − q or x > p + q (q > 0)

"Less than" absolute value inequalities describe a single band of values close to p; "greater than" absolute value inequalities describe two separate regions far from p.

x

— the unknown being solved for

p

— the reference point

q

— the distance threshold (q > 0)

When to use it: Whenever you need to solve an inequality with a single absolute value expression compared to a positive number.

P(E) = n(E) / n(S)

The probability of an event equals the number of favourable outcomes divided by the total number of equally likely outcomes in the sample space.

P(E)

— the probability of event E, a number from 0 to 1

n(E)

— the number of outcomes that make up event E (favourable outcomes)

n(S)

— the total number of outcomes in the sample space S

When to use it: Whenever every outcome in the sample space is equally likely, and you need the probability of a specific event.

P(E') = 1 − P(E)

The probability that an event does NOT happen equals 1 minus the probability that it does happen.

P(E')

— the probability of the complement of E — E not happening

P(E)

— the probability of E happening

When to use it: Whenever finding "the probability that something does not happen" is easier than finding the probability that it does.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A, B are mutually exclusive: P(A ∪ B) = P(A) + P(B).

The probability that A or B happens equals the sum of their individual probabilities, minus the probability they both happen (to avoid double-counting the overlap).

P(A ∪ B)

— the probability that A happens, or B happens, or both

P(A ∩ B)

— the probability that both A and B happen

mutually exclusive

— A and B cannot both happen at once, so P(A ∩ B) = 0

When to use it: Whenever you need the probability that at least one of two events happens.

P(A ∩ B) = P(A) × P(B) (A, B independent)

When two events do not affect each other's outcome, the probability that both happen equals the product of their individual probabilities.

P(A ∩ B)

— the probability that both A and B happen

independent

— the outcome of A has no effect on the probability of B, and vice versa

When to use it: Whenever two events are independent and you need the probability that both occur.

P(A | B) = P(A ∩ B) / P(B) (P(B) ≠ 0)

The probability that A happens, given that B is already known to have happened — restricting the sample space down to just the outcomes where B occurs.

P(A | B)

— the probability of A, given that B has occurred

P(A ∩ B)

— the probability that both A and B happen

P(B)

— the probability that B happens

When to use it: Whenever extra information (that some other event B has already happened) changes the sample space you should be considering.

E = n × P(event)

The number of times an event is predicted to happen if a trial is repeated n times, based on its probability.

E

— the expected frequency — the predicted number of occurrences

n

— the number of trials (repetitions)

P(event)

— the event's probability on a single trial

When to use it: Whenever you need to predict how many times an event will occur over many repeated trials.

AB/A'B' = BC/B'C' = CA/C'A' = k, and corresponding angles are equal

Two triangles are similar exactly when their corresponding angles are equal and their corresponding sides are all in the same fixed ratio, k (the scale factor).

k

— the scale factor — how many times larger (or smaller) one triangle is than the other

AB, BC, CA / A'B', B'C', C'A'

— corresponding side lengths of the two similar triangles

When to use it: Whenever you need to confirm two triangles are similar, or use their known similarity to find a missing side length.

If DE ∥ BC in △ABC (D on AB, E on AC), then AD/DB = AE/EC

A line drawn parallel to one side of a triangle, cutting the other two sides, divides those two sides in exactly the same ratio.

D, E

— the points where the parallel line crosses sides AB and AC

AD, DB, AE, EC

— the four segments the parallel line creates on the two cut sides

When to use it: Whenever a line parallel to one side of a triangle cuts the other two sides, and you need to relate the resulting segment lengths.

If AD bisects ∠A in △ABC (D on BC), then BD/DC = AB/AC

The bisector of an angle of a triangle divides the opposite side into two segments whose ratio equals the ratio of the two sides forming that angle.

D

— the point where the angle bisector from A meets side BC

BD, DC

— the two segments the bisector divides BC into

AB, AC

— the two sides forming the bisected angle at A

When to use it: Whenever an angle bisector in a triangle is given, and you need to find how it divides the opposite side.

a² + b² = c²

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (the legs).

a, b

— the lengths of the two legs (the sides forming the right angle)

c

— the length of the hypotenuse (the side opposite the right angle, always the longest side)

When to use it: Whenever you know two sides of a right triangle and need the third, or need to check whether a triangle is a right triangle.

45-45-90: leg : leg : hypotenuse = x : x : x√2. 30-60-90: short leg : long leg : hypotenuse = x : x√3 : 2x

Two specific right triangles — with angles 45-45-90 and 30-60-90 — always have their side lengths in these exact fixed ratios, letting you find every side from just one known length.

x

— the shortest side (the leg in a 45-45-90 triangle, or the short leg opposite the 30° angle in a 30-60-90 triangle)

When to use it: Whenever a right triangle has angles of 45-45-90 or 30-60-90, letting you find every side instantly from a single known length, without the Pythagorean theorem.

Inscribed angle = (1/2) × central angle, for angles subtending the same arc

An angle inscribed in a circle (with its vertex on the circle) is always exactly half the central angle that subtends the same arc.

central angle

— the angle at the centre of the circle, formed by two radii to the ends of the arc

inscribed angle

— the angle at a point on the circle, formed by two chords to the same ends of the arc

When to use it: Whenever an angle is formed at the centre and another at the circle's edge, both looking at the same arc.

∠A + ∠C = 180° and ∠B + ∠D = 180°

In any quadrilateral whose four vertices all lie on a circle (a cyclic quadrilateral), each pair of opposite angles adds up to exactly 180°.

∠A, ∠B, ∠C, ∠D

— the four interior angles of the cyclic quadrilateral, in order around the circle

When to use it: Whenever a quadrilateral is inscribed in a circle and you need to find a missing angle from its opposite angle.

Chord AB = Chord CD ⇔ AB and CD are equidistant from the centre ⇔ arc AB = arc CD

In the same circle (or equal circles), equal chords are always the same distance from the centre and always cut off equal arcs — and each of these conditions implies the other two.

chord

— a straight segment connecting two points on the circle

distance from the centre

— the length of the perpendicular segment from the centre to the chord

When to use it: Whenever you know one of "equal chords," "equal distance from centre," or "equal arcs," and need to conclude the other two.

If OM ⊥ AB (M on chord AB, O the centre), then AM = MB

A perpendicular line drawn from a circle's centre to any chord always bisects that chord exactly in half.

O

— the centre of the circle

M

— the point where the perpendicular from O meets the chord AB

AM, MB

— the two halves of the chord created by M

When to use it: Whenever a perpendicular is drawn from a circle's centre to a chord, letting you conclude the chord is bisected — often the key step in finding a chord's length from the radius and its distance from the centre.

PA × PC = PB × PD, for chords AC and BD intersecting at point P inside the circle

When two chords intersect inside a circle, the products of the two segments of each chord are always equal.

P

— the point where the two chords intersect

PA, PC

— the two segments the intersection point creates on the first chord

PB, PD

— the two segments the intersection point creates on the second chord

When to use it: Whenever two chords cross inside a circle, and you know three of the four segment lengths and need the fourth.

θ + 360°n (or θ + 2πn radians), for any integer n

Two angles in standard position are coterminal if they share the same terminal side — which happens exactly when they differ by a whole number of full revolutions.

θ

— the original angle

n

— any integer — how many extra full turns are added or subtracted

When to use it: Whenever you need to find an equivalent angle (e.g. within 0°-360°) for an angle greater than 360° or negative.

radians = degrees × (π/180) degrees = radians × (180/π)

Converts an angle measurement between degrees and radians, the two standard units for measuring angles.

π

— pi, approximately 3.14159 — one full revolution is 2π radians, or 360°

When to use it: Whenever an angle is given in one unit but a formula or calculation requires the other.

s = rθ (θ in radians)

The length of an arc equals the radius times the angle it subtends, when that angle is measured in radians.

s

— the arc length

r

— the radius of the circle

θ

— the central angle subtended by the arc, in radians

When to use it: Whenever you need the length of a curved arc from its radius and central angle.

A = (1/2) r² θ (θ in radians)

The area of a pie-slice-shaped sector equals half the radius squared times the central angle, when that angle is in radians.

A

— the area of the sector

r

— the radius of the circle

θ

— the central angle of the sector, in radians

When to use it: Whenever you need the area of a pie-slice-shaped portion of a circle, given its radius and central angle.

sinθ=opp/hyp, cosθ=adj/hyp, tanθ=opp/adj, cscθ=hyp/opp, secθ=hyp/adj, cotθ=adj/opp

For an acute angle θ in a right triangle, each trigonometric ratio is a fixed ratio of two of the triangle's three sides — three basic ratios, and their three reciprocals.

opp

— the length of the side opposite angle θ

adj

— the length of the side adjacent to angle θ (not the hypotenuse)

hyp

— the length of the hypotenuse, opposite the right angle

When to use it: Whenever you know an angle and one side of a right triangle and need another side, or know two sides and need the angle.

sin²θ + cos²θ = 1 also: tanθ = sinθ/cosθ, cotθ = cosθ/sinθ

The squares of sine and cosine of the same angle always add up to exactly 1 — a direct consequence of the Pythagorean theorem — and tangent is always the ratio of sine to cosine.

θ

— any angle

When to use it: Whenever you know one trigonometric ratio of an angle and need another, without knowing the angle itself.

sin: 0, 1/2, √2/2, √3/2, 1 | cos: 1, √3/2, √2/2, 1/2, 0 (for θ = 0°, 30°, 45°, 60°, 90°)

The sine and cosine of the five most common "special" angles have exact, memorizable values, derived directly from the 45-45-90 and 30-60-90 triangles.

θ

— one of the five special angles: 0°, 30°, 45°, 60°, or 90°

When to use it: Whenever a problem involves one of these five common angles and you need an exact value rather than a decimal approximation.

P(x) = D(x) · Q(x) + R(x), where the degree of R(x) is less than the degree of D(x)

Dividing a polynomial P(x) by a divisor D(x) always produces a quotient Q(x) and a remainder R(x), related by this identity — the same "dividend = divisor × quotient + remainder" pattern used for ordinary numbers.

P(x)

— the dividend — the polynomial being divided

D(x)

— the divisor — the polynomial being divided by

Q(x)

— the quotient

R(x)

— the remainder (degree strictly less than D(x), or R(x)=0 if it divides evenly)

When to use it: Whenever you divide one polynomial by another and need to check or express the result correctly.

If P(x) is divided by (x − a), the remainder equals P(a)

Dividing a polynomial by (x − a) always leaves a remainder equal to the polynomial evaluated at x = a — found by a single substitution, without doing the division at all.

P(x)

— the polynomial being divided

a

— the constant in the linear divisor (x − a)

P(a)

— the polynomial evaluated at x = a — this equals the remainder

When to use it: Whenever you need the remainder of a polynomial divided by a linear factor (x − a), without carrying out the full division.

(x − a) is a factor of P(x) ⇔ P(a) = 0

A linear expression (x − a) divides a polynomial P(x) exactly (with zero remainder) exactly when a is a root of P(x) — substituting a into P(x) gives zero.

P(x)

— the polynomial being tested

a

— a candidate root — the constant in the linear factor (x − a)

When to use it: Whenever you need to test whether a specific value is a root of a polynomial, or to find factors of a polynomial by testing candidate values.

ⁿCᵣ = n! / (r!(n − r)!)

Counts how many ways to choose r items from a set of n, when order does not matter — this is exactly the coefficient of each term in a binomial expansion.

n

— the total number of items (or the power in the binomial expansion)

r

— the number of items chosen (or the term index in the expansion)

n!

— n factorial: n × (n−1) × (n−2) × ... × 1

When to use it: Whenever you need a binomial coefficient — either for counting combinations or for the coefficients in a binomial expansion.

(a + b)ⁿ = Σ (from r=0 to n) of ⁿCᵣ · aⁿ⁻ʳ · bʳ

Gives every term of the expansion of (a + b)ⁿ directly, without multiplying out the brackets by hand — each term's coefficient is a binomial coefficient, and the exponents of a and b always add to n.

n

— the power the binomial is raised to

r

— the term index, running from 0 to n

ⁿCᵣ

— the binomial coefficient for that term

When to use it: Whenever you need to expand (a + b)ⁿ for any n, especially when n is too large to expand by repeated multiplication.

T(r+1) = ⁿCᵣ · aⁿ⁻ʳ · bʳ

Gives the formula for the (r+1)th term of the expansion of (a+b)ⁿ directly, without writing out every earlier term first.

T(r+1)

— the (r+1)th term of the expansion (the "+1" exists because the first term corresponds to r=0)

n, r

— as in the binomial theorem — n is the power, r is the term index

When to use it: Whenever you need one specific term of a binomial expansion (e.g. the 5th term, or the term containing x⁴) without expanding the whole thing.

y = ax³ + bx² + cx + d (a ≠ 0)

The general form of a cubic function — the simplest case, y = x³, has a distinctive S-shaped curve that passes through the origin and is symmetric about it (an odd function).

a, b, c, d

— real-number coefficients (a ≠ 0, so the x³ term does not vanish)

When to use it: Whenever you need to recognize or sketch the shape of a cubic function.

y = a / x (x ≠ 0)

A reciprocal function has a two-branch, hyperbola-like curve that approaches (but never touches) both axes — the axes are its asymptotes.

a

— a nonzero constant that scales the curve and determines which two quadrants it occupies

x ≠ 0

— x cannot be zero — the function is undefined there

When to use it: Whenever you need to recognize or sketch the shape of a reciprocal (inverse-proportion) function.

y = a√(x − h) + k (x ≥ h)

A square root function traces half of a sideways parabola, starting at a fixed point (h, k) and extending in only one horizontal direction, since the domain is restricted to x ≥ h.

a

— scales the curve, and flips it if negative

h, k

— the coordinates of the curve's starting point

When to use it: Whenever you need to recognize or sketch the shape of a square root function, and identify its domain restriction.

y = a · bˣ (a ≠ 0, b > 0, b ≠ 1)

An exponential function grows (if b>1) or decays (if 0<b<1) increasingly quickly, and always has a horizontal asymptote at y = 0.

a

— the y-intercept when x=0 (since bˣ⁼⁰ = 1)

b

— the base — determines growth (b>1) or decay (0<b<1)

When to use it: Whenever you need to recognize or sketch the shape of an exponential growth or decay function.

y = f(x − h) + k

Shifts the entire graph of y=f(x) horizontally by h units and vertically by k units, without changing its shape at all.

h

— the horizontal shift (right if h>0, left if h<0 — note the subtraction)

k

— the vertical shift (up if k>0, down if k<0)

When to use it: Whenever a graph needs to be moved to a new location without changing its shape.

y = −f(x) reflects over the x-axis. y = f(−x) reflects over the y-axis.

Negating the whole function output flips the graph upside down (over the x-axis); negating the input flips it left-right (over the y-axis).

f(x)

— the original function

When to use it: Whenever a graph needs to be mirrored over an axis.

y = a·f(x) stretches/compresses vertically. y = f(bx) stretches/compresses horizontally.

Multiplying the output by a scales the graph vertically (|a|>1 stretches, 0<|a|<1 compresses); multiplying the input by b scales it horizontally, but in the opposite sense (|b|>1 compresses, 0<|b|<1 stretches).

a

— the vertical scale factor

b

— the horizontal scale factor (acts inversely — a larger b compresses)

When to use it: Whenever a graph needs to be made taller/shorter or narrower/wider.

aₙ = a₁ + (n − 1)d

Gives the value of any term directly from its position n, without listing every earlier term.

aₙ

— the nth term of the sequence

a₁

— the first term

d

— the common difference — the fixed amount added between consecutive terms

n

— the term's position in the sequence

When to use it: Whenever you need a specific term of an arithmetic sequence directly, or need to find n, a₁, or d from other known values.

Sₙ = n/2 · (a₁ + aₙ) = n/2 · (2a₁ + (n − 1)d)

Gives the sum of the first n terms of an arithmetic sequence directly, without adding every term one at a time.

Sₙ

— the sum of the first n terms

a₁, aₙ

— the first and nth (last) terms being summed

n

— the number of terms

d

— the common difference

When to use it: Whenever you need the total of a run of arithmetic-sequence terms, without adding them individually.

aₙ = a₁ · rⁿ⁻¹

Gives the value of any term of a geometric sequence directly from its position n.

aₙ

— the nth term of the sequence

a₁

— the first term

r

— the common ratio — the fixed factor multiplied between consecutive terms

n

— the term's position in the sequence

When to use it: Whenever you need a specific term of a geometric sequence directly, or need to find n, a₁, or r from other known values.

Sₙ = a₁(1 − rⁿ) / (1 − r) (r ≠ 1)

Gives the sum of the first n terms of a geometric sequence directly, without adding every term.

Sₙ

— the sum of the first n terms

a₁

— the first term

r

— the common ratio (r ≠ 1)

n

— the number of terms being summed

When to use it: Whenever you need the total of a run of geometric-sequence terms, without adding them individually.

S∞ = a₁ / (1 − r) (only valid when |r| < 1)

Gives the finite total that a geometric series with shrinking terms approaches as more and more terms are added, forever.

S∞

— the sum of the infinite series

a₁

— the first term

r

— the common ratio, with |r| < 1 required for the sum to exist

When to use it: Whenever a geometric series continues forever and its common ratio satisfies |r| < 1, so its terms shrink toward zero.

(A ± B)ᵢⱼ = Aᵢⱼ ± Bᵢⱼ

To add or subtract two matrices, add or subtract the entries that sit in the exact same position — this only works when both matrices have identical order (same number of rows and columns).

Aᵢⱼ

— the entry of matrix A in row i, column j

Bᵢⱼ

— the entry of matrix B in the same position

When to use it: Whenever you need to add or subtract two matrices of the same order.

(kA)ᵢⱼ = k · Aᵢⱼ

To multiply a matrix by a scalar (an ordinary number), multiply every single entry of the matrix by that number.

k

— the scalar (a plain number)

Aᵢⱼ

— the entry of matrix A in row i, column j

When to use it: Whenever a whole matrix needs to be scaled up or down by a constant factor.

(AB)ᵢⱼ = Σₖ Aᵢₖ · Bₖⱼ (row i of A "dotted" with column j of B)

Each entry of the product matrix AB comes from multiplying corresponding entries of a row of A with a column of B, and adding the results — matrix multiplication requires the number of columns of A to equal the number of rows of B.

(AB)ᵢⱼ

— the entry of the product AB in row i, column j

Aᵢₖ

— the entries along row i of matrix A

Bₖⱼ

— the entries down column j of matrix B

When to use it: Whenever two compatible matrices (columns of the first = rows of the second) need to be multiplied together.

For A = [[a, b], [c, d]]: det(A) = ad − bc

A single number computed from a 2×2 matrix that reveals whether the matrix has an inverse — a matrix has an inverse exactly when its determinant is nonzero.

a, b, c, d

— the four entries of the 2×2 matrix, read left-to-right, top-to-bottom

det(A)

— the determinant of A, also written |A|

When to use it: Whenever you need to check if a 2×2 matrix has an inverse, or as the first step of computing that inverse.

For A = [[a, b], [c, d]]: A⁻¹ = (1/det(A)) · [[d, −b], [−c, a]] (det(A) ≠ 0)

The inverse of a 2×2 matrix — swap the two diagonal entries, negate the two off-diagonal entries, then divide every entry by the determinant.

A⁻¹

— the inverse of A, satisfying A·A⁻¹ = A⁻¹·A = I (the identity matrix)

det(A)

— the determinant, ad − bc (must be nonzero)

When to use it: Whenever you need to "undo" a 2×2 matrix's effect, or solve a matrix equation of the form AX = B.

Range = maximum − minimum. IQR = Q₃ − Q₁

Two simple measures of how spread out a dataset is. The range uses only the two extreme values; the interquartile range (IQR) uses the middle 50% of the data, making it far less sensitive to outliers.

Q₁

— the first quartile — the value 25% of the way through the ordered data

Q₃

— the third quartile — the value 75% of the way through the ordered data

When to use it: Whenever you need a quick measure of spread — the range for a rough idea, the IQR for a measure resistant to extreme outliers.

Variance: σ² = Σ(x − x̄)² / n. Standard deviation: σ = √(σ²)

Measures how far, on average, the data values are spread from the mean — variance is the average of the squared deviations; standard deviation is its square root, bringing the units back in line with the original data.

x

— each individual data value

x̄

— the mean of the dataset

n

— the number of data values

σ (sigma)

— the standard deviation

When to use it: Whenever you need the standard, most widely-used measure of how spread out a dataset is around its mean.

r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² · Σ(y − ȳ)²]

A single number, always between −1 and 1, that measures both the strength and direction of the linear relationship between two variables.

r

— the correlation coefficient

x, y

— the paired data values for each variable

x̄, ȳ

— the means of the x-values and y-values respectively

When to use it: Whenever you need a precise numerical measure of how strongly two variables are linearly related, beyond just a visual scatter diagram impression.

OT ⊥ the tangent line, at the point of tangency T

A tangent line to a circle is always exactly perpendicular to the radius drawn to the point where it touches the circle.

O

— the centre of the circle

T

— the point of tangency — where the tangent line touches the circle

When to use it: Whenever a tangent line and a radius meet at the point of tangency, guaranteeing a 90° angle there.

PA = PB, for two tangents from external point P touching the circle at A and B

The two tangent segments drawn from any single external point to a circle are always exactly equal in length.

P

— an external point (outside the circle)

A, B

— the two points where the tangents from P touch the circle

When to use it: Whenever two tangent segments are drawn to a circle from the same external point.

The angle between a tangent and a chord (drawn from the point of tangency) equals the inscribed angle in the alternate segment

Also called the Alternate Segment Theorem — the angle formed between a tangent line and a chord equals the angle that same chord subtends from any point on the arc on the OTHER side of the chord.

tangent-chord angle

— the angle between the tangent line and the chord, measured at the point of tangency

alternate segment

— the region of the circle on the opposite side of the chord from the tangent-chord angle being measured

When to use it: Whenever a tangent and a chord meet at a point of tangency, and you need to relate that angle to an inscribed angle elsewhere in the circle.

If ∠ACB = ∠ADB (C, D on the same side of AB), then A, B, C, D are concyclic

If two points, C and D, on the same side of a segment AB, both see AB under the exact same angle, then all four points A, B, C, D must lie on one common circle.

A, B

— two fixed points (forming the reference segment)

C, D

— two points on the same side of AB, each forming an angle with A and B

When to use it: Whenever you need to prove that four points all lie on a single circle, using only angle measurements.

Area₁ / Area₂ = (s₁ / s₂)²

For two similar triangles, the ratio of their areas equals the square of the ratio of any pair of corresponding side lengths.

Area₁, Area₂

— the areas of the two similar triangles

s₁, s₂

— a pair of corresponding side lengths (or the linear scale factor between the two triangles), one from each triangle

When to use it: Whenever you know (or can find) a pair of corresponding side lengths of two similar triangles and need to relate their areas.

Perimeter₁ / Perimeter₂ = s₁ / s₂

For two similar triangles, the ratio of their perimeters equals the ratio of any pair of corresponding side lengths directly — the same linear scale factor, not squared.

Perimeter₁, Perimeter₂

— the perimeters of the two similar triangles

s₁, s₂

— a pair of corresponding side lengths, one from each triangle

When to use it: Whenever you need to relate the perimeters (not the areas) of two similar triangles to their scale factor.

AB + BC = AC

Placing the tail of a second vector at the head of the first, the sum is the single vector from the very first tail to the very last head.

AB

— a vector from point A to point B

BC

— a vector from point B to point C, placed tail-to-head with AB

AC

— the resultant (sum) vector, from A directly to C

When to use it: Whenever two vectors are added and can be drawn head-to-tail (e.g. combining two journeys, or two forces acting in sequence).

a − b = a + (−b)

Subtracting a vector b is the same as adding its negative, −b — the same vector as b but reversed in direction, with the same magnitude.

a, b

— the two vectors being subtracted

−b

— the vector with the same magnitude as b, pointing the opposite way

When to use it: Whenever a vector must be subtracted from another — reverse the second vector's direction, then add.

k·a has magnitude |k||a|, same direction as a if k>0, opposite if k<0

Multiplying a vector by a scalar (an ordinary number) k stretches or shrinks its magnitude by a factor of |k|, and flips its direction if k is negative.

k

— a scalar (real number) multiplier

a

— the original vector

|a|

— the magnitude (length) of vector a

When to use it: Whenever a vector needs to be scaled in size, reversed in direction, or both.

If P divides AB in the ratio m:n, then OP = (n·OA + m·OB) / (m+n)

Gives the position vector of a point P that divides the segment AB in a given ratio m:n, directly from the position vectors of A and B.

O

— the fixed origin

OA, OB, OP

— the position vectors of A, B, and P respectively

m : n

— the ratio in which P divides AB, measured from A to B

When to use it: Whenever a point divides a line segment in a known ratio and you need that point's position vector (e.g. the midpoint is the special case m=n=1).

AB = OB − OA

The vector from point A to point B equals B's position vector minus A's position vector.

O

— the fixed origin

OA, OB

— the position vectors of A and B

AB

— the vector from A to B

When to use it: Whenever you know the position vectors of two points and need the vector that goes from one to the other.

|v| = √(x² + y²), for v = (x, y)

The magnitude (length) of a two-dimensional vector is found by applying the Pythagorean theorem to its horizontal and vertical components.

v

— a two-dimensional vector, written in column/component form

x, y

— the horizontal and vertical components of v

|v|

— the magnitude (length) of v

When to use it: Whenever a vector is given in component (column) form and you need its actual length.

sin θ = y/r, cos θ = x/r, tan θ = y/x

For any angle θ (not just acute angles), place its vertex at the origin with one ray along the positive x-axis; take any point (x, y) on the other ray at distance r from the origin, and define the ratios from those coordinates.

θ

— the angle, measured counterclockwise from the positive x-axis

(x, y)

— any point on the angle's terminal ray

r

— the distance from the origin to (x, y); always positive, r = √(x²+y²)

When to use it: Whenever a trigonometric ratio is needed for an angle that is not between 0° and 90°.

Quadrant I: All positive. II: Sin (and cosec) positive. III: Tan (and cot) positive. IV: Cos (and sec) positive.

A quick way to remember which trigonometric ratio is positive in each quadrant — often remembered with a phrase like "All Students Take Calculus," reading the quadrants in order I, II, III, IV.

Quadrant I

— 0° to 90° — x and y both positive

Quadrant II

— 90° to 180° — x negative, y positive

Quadrant III

— 180° to 270° — x and y both negative

Quadrant IV

— 270° to 360° — x positive, y negative

When to use it: Whenever you need to determine whether a trigonometric ratio should be positive or negative for a given angle.

sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ

A negative angle is measured clockwise instead of counterclockwise. Sine and tangent of a negative angle are the negative of the positive angle's value; cosine is unchanged.

−θ

— the angle θ measured in the clockwise direction instead

When to use it: Whenever an angle is given as negative (measured clockwise), to rewrite it in terms of the equivalent positive angle's ratios.

Quadrant II: α = 180°−θ. Quadrant III: α = θ−180°. Quadrant IV: α = 360°−θ

The basic (reference) acute angle α is the acute angle formed between the terminal ray of θ and the x-axis — it lets any trig ratio be evaluated using known acute-angle values, combined with the correct quadrant sign.

θ

— the original angle, in any quadrant

α

— the basic acute angle, always between 0° and 90°

When to use it: Whenever a trigonometric ratio is needed for an angle greater than 90°, to reduce the problem to a familiar acute-angle calculation.

sin: 0, 1, 0, −1, 0. cos: 1, 0, −1, 0, 1. tan: 0, undefined, 0, undefined, 0 — at 0°, 90°, 180°, 270°, 360°

The sine, cosine, and tangent values at the four axis-aligned "quadrantal" angles (and back to 0°/360°), found by taking a point directly on an axis rather than inside a quadrant.

0°, 360°

— point on the positive x-axis: (x,y) = (r, 0)

90°

— point on the positive y-axis: (x,y) = (0, r)

180°

— point on the negative x-axis: (x,y) = (−r, 0)

270°

— point on the negative y-axis: (x,y) = (0, −r)

When to use it: Whenever a trigonometric ratio is needed for exactly 0°, 90°, 180°, 270°, or 360° — including recognizing where tan is undefined.

sin²θ + cos²θ = 1

For any angle θ, the square of its sine plus the square of its cosine always equals exactly 1.

θ

— any angle

When to use it: Whenever one of sin θ or cos θ is known and the other is needed, or to simplify an expression containing both.

tan θ = sin θ / cos θ

The tangent of an angle is always equal to its sine divided by its cosine.

θ

— any angle for which cos θ ≠ 0

When to use it: Whenever tan θ needs to be found from known sin θ and cos θ values, or vice versa.

a/sin A = b/sin B = c/sin C

In any triangle, the ratio of a side's length to the sine of its opposite angle is the same for all three sides.

a, b, c

— the side lengths of the triangle

A, B, C

— the angles opposite sides a, b, c respectively

When to use it: Whenever you know two angles and one side (AAS/ASA), or two sides and a non-included angle (SSA), of any triangle (not just right triangles).

c² = a² + b² − 2ab·cos C

A generalization of the Pythagorean theorem to any triangle — gives the length of a side from the other two sides and the angle between them, or an angle from all three sides.

a, b, c

— the side lengths of the triangle

C

— the angle opposite side c, included between sides a and b

When to use it: Whenever you know two sides and the included angle (SAS), or all three sides (SSS), of any triangle.

Area = ½ab·sin C

The area of any triangle can be found from two sides and the sine of the angle between them, without needing to know the height directly.

a, b

— two side lengths of the triangle

C

— the angle included between sides a and b

When to use it: Whenever two sides and the included angle of a triangle are known, and the height is not directly given.

Average rate of change = [f(b) − f(a)] / (b − a)

The average rate of change of a function f between x=a and x=b is the slope of the secant line connecting the two points (a, f(a)) and (b, f(b)).

a, b

— the two x-values marking the interval

f(a), f(b)

— the function's values at those two x-values

When to use it: Whenever you need the average rate of change of a function over an interval, before narrowing that interval down to an instantaneous rate.

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

The derivative of f at x is the limit of the average rate of change over a shrinking interval [x, x+h], as that interval's width h shrinks to zero — giving the exact instantaneous rate of change.

h

— a small change in x, which shrinks toward (but never reaches) 0

f'(x)

— the derivative of f at x — the instantaneous rate of change, and the slope of the tangent line there

When to use it: Whenever a derivative needs to be found directly from its formal limit definition (rather than a shortcut rule).

d/dx[x^n] = n·x^(n−1), d/dx[c·f(x)] = c·f'(x)

To differentiate a power of x, bring the exponent down as a multiplying factor and reduce the exponent by 1. A constant multiplying a function simply carries through unchanged.

n

— any real-number exponent

c

— a constant multiplier

When to use it: Whenever differentiating a term of the form x^n, or a constant times a function.

d/dx[f(x) ± g(x)] = f'(x) ± g'(x)

The derivative of a sum (or difference) of functions is simply the sum (or difference) of their individual derivatives.

f(x), g(x)

— two differentiable functions

When to use it: Whenever a function is written as terms added or subtracted together — differentiate each term separately.

(f·g)' = f'·g + f·g'

The derivative of a product of two functions is the derivative of the first times the second, plus the first times the derivative of the second.

f, g

— two differentiable functions of x

When to use it: Whenever two functions of x are multiplied together — never just multiply their individual derivatives.

(f/g)' = (f'·g − f·g') / g²

The derivative of a quotient of two functions follows a specific pattern: (derivative of top × bottom, minus top × derivative of bottom), all divided by the bottom squared.

f, g

— two differentiable functions of x, with g(x) ≠ 0

When to use it: Whenever one function of x is divided by another.

d/dx[f(g(x))] = f'(g(x))·g'(x)

To differentiate a composite function (a function of a function), differentiate the outer function first (leaving the inner function unchanged inside it), then multiply by the derivative of the inner function.

f(g(x))

— a composite function — f applied to g(x)

g'(x)

— the derivative of the inner function g

When to use it: Whenever differentiating a function that is "wrapped around" another function, e.g. (3x+1)^5 or sin(x²).

Differentiate both sides w.r.t. x; for every y-term, multiply by dy/dx (Chain Rule)

When y is not isolated on one side of an equation, differentiate both sides of the equation with respect to x directly, treating y as an unknown function of x — every time a y appears, its derivative contributes a dy/dx factor by the Chain Rule.

dy/dx

— the derivative of y with respect to x — the very thing being solved for

When to use it: Whenever an equation relates x and y together, but y cannot easily be isolated as y = (some function of x).

i² = −1 (powers of i cycle: i, −1, −i, 1, i, −1, ...)

i is defined so that squaring it gives −1. Every higher power of i repeats this same cycle of four values.

i

— the imaginary unit, defined by i² = −1

When to use it: Whenever i² (or any higher power of i) appears in an expression and needs to be simplified.

z = a + bi

Every complex number can be written as a real number a plus a real number b times i — a is the real part, b is the imaginary part.

a

— the real part of z, written Re(z)

b

— the imaginary part of z, written Im(z) — a real number, the coefficient of i

When to use it: Whenever a complex number needs to be identified or written in its most basic, universal form.

(a+bi) ± (c+di) = (a±c) + (b±d)i

Combine the two real parts together, and separately combine the two imaginary parts together.

a+bi, c+di

— the two complex numbers being added or subtracted

When to use it: Whenever two complex numbers are added or subtracted.

(a+bi)(c+di) = (ac−bd) + (ad+bc)i

Expand using the distributive property (like multiplying two binomials), then replace i² with −1 and combine like terms.

a+bi, c+di

— the two complex numbers being multiplied

When to use it: Whenever two complex numbers are multiplied together.

(a+bi) / (c+di) = [(a+bi)(c−di)] / (c²+d²)

Multiply both the numerator and denominator by the denominator's conjugate (c−di) — this makes the denominator a real number (c²+d²), clearing i from it entirely.

c−di

— the complex conjugate of the denominator c+di — same real part, opposite sign on the imaginary part

c²+d²

— the result of (c+di)(c−di), always a real, non-negative number

When to use it: Whenever dividing by a complex number.

|z| = √(a² + b²), for z = a + bi

The modulus is the distance from the origin to the point representing z on the complex plane — found using the Pythagorean theorem on the real and imaginary parts.

a, b

— the real and imaginary parts of z

|z|

— the modulus (magnitude) of z, always a non-negative real number

When to use it: Whenever the size (magnitude) of a complex number is needed, independent of its direction.

z = r(cos θ + i sin θ), where r = |z| and θ = arg(z)

Any complex number can be written using its modulus r and its argument (angle) θ, instead of its real and imaginary parts directly.

r

— the modulus of z, r = √(a²+b²)

θ

— the argument of z — the angle the line to z makes with the positive real axis

When to use it: Whenever a complex number needs to be expressed by its size and direction, especially before multiplying/dividing or raising to a power.

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ))

Raising a complex number in trigonometric form to a power n raises its modulus to the power n, and multiplies its argument by n.

n

— the power the complex number is being raised to

When to use it: Whenever a complex number needs to be raised to a power, especially a large one — far faster than repeated multiplication.

zₖ = r^(1/n) [cos((θ+360°k)/n) + i sin((θ+360°k)/n)], for k = 0, 1, ..., n−1

Every nonzero complex number has exactly n distinct nth roots, all with the same modulus r^(1/n), evenly spaced 360°/n apart around a circle.

k

— an integer from 0 to n−1, giving one distinct root each

r^(1/n)

— the (positive, real) nth root of the modulus — the same for every root

When to use it: Whenever all n solutions to zⁿ = (some complex number) are needed.

If P(1) is true, and P(k) true ⟹ P(k+1) true for every k, then P(n) is true for all n ≥ 1

A two-step method for proving a statement P(n) holds for every integer n from some starting value onward, without checking each one individually.

P(n)

— the mathematical statement being proved, depending on the integer n

Base case

— the direct verification that P(1) (or the actual starting value) is true

Inductive step

— proving that IF P(k) is true, THEN P(k+1) must also be true

When to use it: Whenever a statement needs to be proved true for every integer in an infinite range, not just checked for a handful of examples.

1 + 2 + 3 + ... + n = n(n+1) / 2

The sum of every positive integer from 1 up to n, provable by mathematical induction.

n

— the largest integer included in the sum, a positive integer

When to use it: Whenever the sum of consecutive integers from 1 to n is needed directly, without adding them one by one.

Assume f(k) = d·m for some integer m; show f(k+1) = d·(some integer)

To prove f(n) is always divisible by a fixed number d, verify the base case directly, then assume f(k) equals d times some integer, and algebraically show f(k+1) must also equal d times some integer.

d

— the fixed divisor being proved to always divide f(n)

m

— the (unknown but existing) integer such that f(k) = d·m, by the inductive hypothesis

When to use it: Whenever a formula or expression needs to be proved divisible by a fixed number for every positive integer n.

Assume P(k) [an inequality] holds; use it, plus algebra, to show P(k+1) holds too

To prove an inequality holds for every integer n (from some starting value), verify the base case directly, then use the inductive hypothesis (the inequality at k) to derive the inequality at k+1.

Starting value

— the base case does not have to be n=1 — it can be any integer where the inequality first becomes true

When to use it: Whenever an inequality (rather than an equation) needs to be proved true for every integer in a range.

d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²], M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)

Both the 2D distance and midpoint formulas extend directly to 3D, simply by including a third (z) term.

(x₁,y₁,z₁), (x₂,y₂,z₂)

— the coordinates of the two points

M

— the midpoint of the segment joining the two points

When to use it: Whenever the distance between two points in 3D space, or the point exactly halfway between them, is needed.

a = x₂−x₁, b = y₂−y₁, c = z₂−z₁

Given two points on a line, the direction numbers (a,b,c) describe the line's direction — proportional to how far apart the two points are along each axis.

(x₁,y₁,z₁), (x₂,y₂,z₂)

— any two distinct points on the line

a, b, c

— the direction numbers of the line

When to use it: Whenever the direction of a line in 3D needs to be described, before writing its equation.

(x−x₀)/a = (y−y₀)/b = (z−z₀)/c (symmetric); x=x₀+at, y=y₀+bt, z=z₀+ct (parametric)

A line through a known point (x₀,y₀,z₀) with direction numbers (a,b,c) can be written either as one combined symmetric equation, or as three separate parametric equations depending on a parameter t.

(x₀,y₀,z₀)

— a known point on the line

a, b, c

— the direction numbers of the line

t

— the parameter — each value of t gives one point on the line

When to use it: Whenever a line in 3D needs to be written as an equation, from a known point and direction.

a(x−x₀) + b(y−y₀) + c(z−z₀) = 0, equivalently ax+by+cz = d

A plane through a known point (x₀,y₀,z₀), perpendicular to a normal vector (a,b,c), is described by this equation — the normal vector's components become the plane equation's coefficients.

(a,b,c)

— the normal vector — perpendicular to every line lying in the plane

(x₀,y₀,z₀)

— a known point on the plane

d

— the constant ax₀+by₀+cz₀, once the equation is expanded

When to use it: Whenever a plane needs to be described by an equation, from a known point and its normal direction.

distance = |ax₀+by₀+cz₀−d| / √(a²+b²+c²)

The shortest (perpendicular) distance from a point (x₀,y₀,z₀) to a plane ax+by+cz=d.

(x₀,y₀,z₀)

— the point whose distance to the plane is being found

a, b, c, d

— the coefficients of the plane's equation, ax+by+cz=d

When to use it: Whenever the shortest distance from a point to a plane is needed.

(x−h)² + (y−k)² + (z−l)² = r²

A sphere is the set of all points at a fixed distance r (the radius) from a fixed center point (h,k,l).

(h,k,l)

— the center of the sphere

r

— the radius of the sphere

When to use it: Whenever a sphere needs to be described by an equation, or its center and radius identified from one.

|v| = √(x² + y² + z²), for v = (x, y, z)

The length of a 3D vector, found by extending the Pythagorean theorem to a third dimension.

v

— a three-dimensional vector

x, y, z

— the components of v

When to use it: Whenever the actual length of a 3D vector is needed.

a · b = a₁b₁ + a₂b₂ + a₃b₃

The dot (scalar) product of two vectors is found by multiplying corresponding components and adding the results — the output is always a single number (a scalar), not a vector.

a = (a₁,a₂,a₃), b = (b₁,b₂,b₃)

— the two vectors being combined

When to use it: Whenever two vectors need to be combined into a single scalar value, especially to find the angle between them.

cos θ = (a · b) / (|a||b|)

The angle between two vectors can be found from their dot product, divided by the product of their magnitudes.

θ

— the angle between vectors a and b

|a|, |b|

— the magnitudes of the two vectors

When to use it: Whenever the angle between two vectors needs to be found.

a × b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁)

The cross (vector) product of two vectors produces a NEW vector, perpendicular to both original vectors.

a = (a₁,a₂,a₃), b = (b₁,b₂,b₃)

— the two vectors being combined

a × b

— the resulting vector, perpendicular to both a and b

When to use it: Whenever a vector perpendicular to two given vectors is needed.

Area = |a × b|

The area of the parallelogram formed by two vectors a and b equals the magnitude of their cross product.

a, b

— the two vectors forming the sides of the parallelogram

When to use it: Whenever the area of a parallelogram (or, halved, a triangle) formed by two vectors is needed.

r = r₀ + t·v

A line through the point with position vector r₀, in the direction of vector v, is described by this equation — every value of t gives a different point on the line.

r

— the position vector of a general point on the line

r₀

— the position vector of a known point on the line

v

— the direction vector of the line

t

— the parameter

When to use it: Whenever a line in 3D needs to be described using vector notation.

n · (r − r₀) = 0

A plane through the point with position vector r₀, with normal vector n, is described by this equation — it says every point r in the plane forms a vector (r−r₀) perpendicular to n.

n

— the normal vector to the plane

r₀

— the position vector of a known point on the plane

r

— the position vector of a general point on the plane

When to use it: Whenever a plane needs to be described using vector notation.

Volume = |a · (b × c)|

The volume of the parallelepiped (a 3D "slanted box") formed by three vectors a, b, and c, sharing a common vertex.

a, b, c

— the three edge vectors of the parallelepiped, from one shared corner

When to use it: Whenever the volume of a parallelepiped formed by three vectors is needed.

If one task can be done m ways and a second task n ways, both together = m × n ways

When a choice involves multiple independent stages, the total number of possible outcomes is the product of the number of options at each stage.

m, n

— the number of ways to complete each independent stage of the choice

When to use it: Whenever a total count is needed for a process made up of several independent stages, each with its own number of options.

n! = n × (n−1) × (n−2) × ... × 2 × 1, with 0! = 1 by definition

The factorial of a positive integer n is the product of every positive integer up to n — it counts the number of ways to arrange n distinct objects in a row.

n

— a non-negative integer

When to use it: Whenever the number of full arrangements of a set of distinct objects is needed, or as a building block in permutation and combination formulas.

P(n,n) = n!

The number of different orders in which all n distinct objects can be arranged.

n

— the total number of distinct objects being arranged

When to use it: Whenever every object in a set is being arranged in some order (not just a subset of them).

P(n,r) = n! / (n−r)!

The number of ways to choose AND arrange r objects, in order, out of a total of n distinct objects.

n

— the total number of distinct objects available

r

— the number of objects being chosen and arranged, r ≤ n

When to use it: Whenever the ORDER of the chosen objects matters (e.g. 1st, 2nd, 3rd place).

n! / (n₁! × n₂! × ... × nₖ!)

The number of distinct arrangements of n objects, when some of them are indistinguishable copies of each other, grouped into k types.

n

— the total number of objects (including repeats)

n₁, n₂, ..., nₖ

— the number of repeated copies within each distinct type

When to use it: Whenever arranging a set that includes repeated (indistinguishable) items, like letters in a word with repeated letters.

(n−1)!

The number of distinct ways to arrange n distinct objects around a circle, where rotations of the same arrangement are considered identical.

n

— the number of distinct objects arranged around the circle

When to use it: Whenever objects are arranged in a circle (like people around a table) rather than in a row.

C(n,r) = n! / [r!(n−r)!]

The number of ways to choose r objects from a total of n distinct objects, where the ORDER of selection does not matter.

n

— the total number of distinct objects available

r

— the number of objects being chosen, r ≤ n

When to use it: Whenever a group or subset is being chosen and the order within the group does not matter (like choosing a committee).

P(n,r) = C(n,r) × r!

The number of ordered arrangements (permutations) of r objects equals the number of unordered selections (combinations) of the same r objects, multiplied by the r! ways to order each selected group.

P(n,r)

— the number of ordered arrangements of r objects from n

C(n,r)

— the number of unordered selections of r objects from n

When to use it: Whenever converting between a permutation count and the corresponding combination count (or vice versa).

(x−h)² + (y−k)² = r²

A circle is the set of all points at a fixed distance r (the radius) from a fixed center (h,k).

(h,k)

— the center of the circle

r

— the radius of the circle

When to use it: Whenever a circle needs to be described by an equation, or its center and radius read off directly.

x² + y² + Dx + Ey + F = 0

A circle's equation, expanded out rather than in standard center-radius form — completing the square on the x and y terms recovers the standard form.

D, E, F

— the expanded equation's coefficients

When to use it: Whenever a circle's equation is given in expanded form and its center/radius need to be found.

distance to focus = distance to directrix, for every point on the parabola

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

focus

— a fixed point inside the curve of the parabola

directrix

— a fixed line on the outside of the curve, the same perpendicular distance rule applies to

When to use it: Whenever a parabola needs to be defined or verified from its geometric (rather than algebraic) definition.

x² = 4py, with focus (0,p) and directrix y=−p

A parabola with vertex at the origin, opening upward (p>0) or downward (p<0), described directly by this equation.

p

— the distance from the vertex to the focus (and also to the directrix)

When to use it: Whenever a parabola's vertex is at the origin and its focus/directrix need to be found from the equation, or vice versa.

(x−h)² = 4p(y−k), vertex at (h,k)

A parabola whose vertex has been moved away from the origin, to the point (h,k) — the direct translation of the standard-form equation.

(h,k)

— the vertex of the parabola

p

— the distance from the vertex to the focus

When to use it: Whenever a parabola's vertex is NOT at the origin.

x' = x−h, y' = y−k (new origin at (h,k) in the old system)

Converts a point's coordinates from an original (x,y) system to a new (x',y') system whose origin has been shifted to the point (h,k).

(h,k)

— the new origin's location, measured in the old coordinate system

(x',y')

— the point's coordinates in the new, shifted system

When to use it: Whenever it is more convenient to re-measure coordinates relative to a new, shifted origin.

x = x'cosθ − y'sinθ, y = x'sinθ + y'cosθ

Relates a point's coordinates in an original (x,y) system to its coordinates in a new (x',y') system, rotated by angle θ about the same origin.

θ

— the angle the new axes are rotated by, relative to the old axes

When to use it: Whenever the coordinate axes need to be rotated to align with a tilted feature of a problem, simplifying its equation.

Domain: all real numbers. Range: [−1, 1]. Period: 2π

y=sinx oscillates smoothly between −1 and 1, repeating its exact shape every 2π, and passes through the origin.

x

— the input angle, in radians

When to use it: Whenever identifying or sketching the basic sine graph and its key features.

Domain: all real numbers. Range: [−1, 1]. Period: 2π

y=cosx oscillates smoothly between −1 and 1, repeating every 2π, starting at its maximum value (1) when x=0.

x

— the input angle, in radians

When to use it: Whenever identifying or sketching the basic cosine graph and its key features.

Domain: all x except π/2+kπ. Range: all real numbers. Period: π

y=tanx has vertical asymptotes wherever cosx=0, is unbounded (covers all real numbers), and repeats every π — half the period of sine and cosine.

k

— any integer, giving the full set of excluded x-values

When to use it: Whenever identifying or sketching the tangent graph, especially its asymptotes.

csc x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x

Cosecant, secant, and cotangent are defined as the reciprocals of sine, cosine, and tangent respectively.

csc, sec, cot

— the reciprocal trigonometric functions

When to use it: Whenever cosecant, secant, or cotangent needs to be evaluated or graphed.

y = A sin(B(x−C)) + D

A general transformed sine wave, with amplitude |A|, period 2π/B, phase shift C, and vertical shift D.

A

— controls the amplitude (height) of the wave

B

— controls the period, via period = 2π/B

C

— the horizontal (phase) shift

D

— the vertical shift

When to use it: Whenever a sine (or cosine) graph has been stretched, compressed, or shifted from its basic form.

y = −sin x (reflection over the x-axis); y = sin x + D (vertical shift by D)

A negative sign in front of a trig function reflects its graph over the x-axis; adding a constant D shifts the entire graph up (or down, if D is negative) by D units.

D

— the amount the graph is shifted vertically

When to use it: Whenever a trig graph has been flipped or shifted up/down from its basic form.

y = arcsin(x), domain [−1,1], range [−π/2, π/2]

arcsin(x) gives the ONE angle, within the restricted range [−π/2,π/2], whose sine equals x.

x

— a value between −1 and 1

When to use it: Whenever an angle needs to be recovered from a known sine value.

arccos(x): domain [−1,1], range [0,π]. arctan(x): domain all reals, range (−π/2,π/2)

arccos(x) gives the one angle in [0,π] whose cosine is x; arctan(x) gives the one angle in (−π/2,π/2) whose tangent is x.

x

— the known cosine value (for arccos) or tangent value (for arctan)

When to use it: Whenever an angle needs to be recovered from a known cosine or tangent value.

d/dx[sin x] = cos x, d/dx[cos x] = −sin x

The derivative of sine is cosine; the derivative of cosine is negative sine.

x

— the input angle, in radians

When to use it: Whenever differentiating an expression involving sine or cosine.

d/dx[tan x]=sec²x, d/dx[cot x]=−csc²x, d/dx[sec x]=sec x·tan x, d/dx[csc x]=−csc x·cot x

The derivatives of the remaining four trigonometric functions, each expressible using the reciprocal trig functions.

x

— the input angle, in radians

When to use it: Whenever differentiating an expression involving tangent, cotangent, secant, or cosecant.

y = bˣ (b>0, b≠1): domain all reals, range (0,∞), y-intercept (0,1), asymptote y=0

An exponential function grows (or decays, if 0<b<1) by a constant multiplicative factor for every unit increase in x, and is always strictly positive.

b

— the base of the exponential function, a positive number not equal to 1

When to use it: Whenever identifying or sketching the basic exponential graph and its key features.

y = log_b(x) (b>0, b≠1): domain (0,∞), range all reals, x-intercept (1,0), asymptote x=0

A logarithmic function is the inverse of the exponential function with the same base — it is only defined for positive inputs, and can output any real number.

b

— the base of the logarithm, a positive number not equal to 1

When to use it: Whenever identifying or sketching the basic logarithmic graph and its key features.

y = A·bˣ⁻ᶜ + D (exponential); y = A·log_b(x−C) + D (logarithmic)

Shifting, stretching, or reflecting an exponential or logarithmic graph works the same way as for any other function — C shifts horizontally, D shifts vertically (moving the asymptote along with it), and A stretches/reflects.

A

— a vertical stretch/reflection factor

C

— the horizontal shift

D

— the vertical shift — also shifts the asymptote

When to use it: Whenever an exponential or logarithmic graph has been shifted or scaled from its basic form.

d/dx[eˣ] = eˣ

The exponential function with base e is famously its own derivative — the rate of change of eˣ, at any point, equals its own value there.

e

— Euler's number, approximately 2.71828

When to use it: Whenever differentiating an expression involving eˣ.

d/dx[bˣ] = bˣ · ln(b)

For any base b, the derivative of bˣ is bˣ multiplied by an extra constant factor, ln(b) — this reduces to the simple eˣ case exactly when b=e, since ln(e)=1.

b

— the base of the exponential function

ln(b)

— the natural logarithm of the base

When to use it: Whenever differentiating an exponential function with a base other than e.

d/dx[ln x] = 1/x

The derivative of the natural logarithm is simply the reciprocal of x.

x

— a positive real number (the domain of ln x)

When to use it: Whenever differentiating an expression involving ln x.

d/dx[log_b x] = 1 / (x · ln b)

For any base b, the derivative of log_b(x) includes an extra factor of 1/ln(b) — this reduces to the simple 1/x case exactly when b=e, since ln(e)=1.

b

— the base of the logarithm

When to use it: Whenever differentiating a logarithmic function with a base other than e.

A critical point of f occurs where f'(x) = 0, or where f'(x) is undefined

Critical points are the candidate locations for local maxima, local minima, or other notable features of a function's graph.

f'(x)

— the derivative of f

When to use it: Whenever locating a function's local extrema (maxima/minima) is the goal — always start by finding the critical points.

f'(x) > 0 ⟹ f is increasing; f'(x) < 0 ⟹ f is decreasing

The sign of the derivative directly reveals whether the original function is rising or falling at that point.

f'(x)

— the derivative of f, evaluated at a specific x

When to use it: Whenever the intervals where a function increases or decreases need to be identified.

f' changes + to − at c ⟹ local max at c; f' changes − to + at c ⟹ local min at c

Examining how the derivative's sign changes on either side of a critical point classifies it as a local maximum, local minimum, or neither.

c

— a critical point being classified

When to use it: Whenever a critical point needs to be classified as a local max, local min, or neither — and whenever absolute extrema on a closed interval are needed (also checking the endpoints).

f''(x) > 0 ⟹ concave up; f''(x) < 0 ⟹ concave down

The sign of the second derivative reveals whether the graph curves upward (like a cup) or downward (like a frown) at that point. A point where concavity changes is an inflection point.

f''(x)

— the second derivative of f — the derivative of the derivative

When to use it: Whenever the curvature (concavity) of a graph, or its inflection points, need to be identified.

At a critical point c where f'(c)=0: f''(c)>0 ⟹ local min; f''(c)<0 ⟹ local max; f''(c)=0 ⟹ inconclusive

An alternative (often faster) way to classify a critical point, using the sign of the second derivative there instead of checking the sign change of the first derivative on both sides.

c

— a critical point where f'(c)=0

When to use it: Whenever a critical point needs to be classified, and computing the second derivative is more convenient than the First Derivative Test.

L(x) = f(a) + f'(a)(x−a)

Uses the tangent line at a known point a to estimate the value of f at a nearby point x — since the tangent line closely hugs the curve near the point of tangency.

a

— a known point where f(a) and f'(a) can be computed exactly

x

— a nearby point where f(x) is being estimated

L(x)

— the linear approximation (estimate) of f(x)

When to use it: Whenever a quick estimate of a function's value is needed near a point where the exact value is already known.

∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)

Integration is the reverse of differentiation — this rule finds the family of functions whose derivative is xⁿ, up to an unknown constant C.

n

— the exponent on x (any real number except −1)

C

— the constant of integration — any constant has a derivative of 0, so it cannot be recovered from the derivative alone

When to use it: Whenever integrating a single power of x, or any polynomial (by applying the rule to each term).

∫eˣ dx = eˣ + C; ∫sin(x) dx = −cos(x) + C; ∫cos(x) dx = sin(x) + C

The standard antiderivatives for the exponential and core trigonometric functions, each the reverse of a standard derivative rule.

C

— the constant of integration

When to use it: Whenever integrating a term that is exactly eˣ, sin(x), or cos(x) (or a constant multiple of one).

∫f(g(x))·g'(x) dx = ∫f(u) du, where u = g(x)

Replaces a complicated integral in x with a simpler one in a new variable u, by recognizing the integrand as the result of a Chain Rule differentiation.

u

— the substituted "inner function," u = g(x)

du

— du = g'(x) dx — the differential of u, which must also appear in the integrand

When to use it: Whenever the integrand contains a function and (a constant multiple of) that function's own derivative multiplying it.

∫u dv = uv − ∫v du

Reverses the Product Rule for differentiation, turning one hard integral (∫u dv) into a simpler one (∫v du) plus an easy term (uv).

u

— the factor chosen to be differentiated (should get simpler when differentiated)

dv

— the factor chosen to be integrated (should be easy to integrate)

When to use it: Whenever the integrand is a product of two different types of functions (e.g. a polynomial times an exponential, or a polynomial times a trigonometric function) that substitution cannot simplify.

(px+q) / [(x−a)(x−b)] = A/(x−a) + B/(x−b)

Splits a rational function with two distinct linear factors in its denominator into a sum of simpler fractions, each easy to integrate on its own.

A, B

— constants found by clearing denominators and substituting x=a and x=b

a, b

— the two distinct roots of the denominator

When to use it: Whenever a rational function needs to be integrated and its denominator factors into distinct linear factors.

∫[A/(x−a) + B/(x−b)] dx = A·ln|x−a| + B·ln|x−b| + C

Once a rational function is decomposed into simple fractions of the form A/(x−a), each one integrates directly to a natural logarithm term.

A, B

— the constants found during the decomposition step

C

— the constant of integration

When to use it: As the final step after decomposing a rational function into partial fractions.

∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x)

Connects the two central operations of calculus — differentiation and integration — by showing that a definite integral can be evaluated using any antiderivative, without needing to sum infinitely many slices directly.

a, b

— the lower and upper limits of integration

F

— any antiderivative of f (F'(x)=f(x))

When to use it: Whenever a definite integral needs to be evaluated exactly, given a function with a known antiderivative.

Area = ∫ₐᵇ f(x) dx (for f(x) ≥ 0 on [a,b])

The definite integral of a non-negative function over [a,b] equals the exact area trapped between the curve and the x-axis over that interval.

f(x)

— the function whose graph forms the top boundary of the region

a, b

— the left and right x-boundaries of the region

When to use it: Whenever the exact area between a curve and the x-axis, over a specific interval, is needed.

Area = ∫ₐᵇ [f(x) − g(x)] dx (where f(x) ≥ g(x) on [a,b])

The area of the region trapped between two curves equals the integral of the "top" function minus the "bottom" function, over the interval where the top function stays on top.

f(x)

— the upper (greater-valued) function on [a,b]

g(x)

— the lower (lesser-valued) function on [a,b]

a, b

— the x-values where the two curves intersect (or the given boundaries)

When to use it: Whenever the area of a region enclosed between two curves is needed — first find where they intersect, then integrate their difference.

V = π ∫ₐᵇ [f(x)]² dx

Rotating the region under a curve f(x) around the x-axis sweeps out a solid; slicing it into thin circular disks of radius f(x) and summing their volumes (via integration) gives the total volume.

f(x)

— the radius of the disk at position x — the distance from the x-axis to the curve

a, b

— the x-boundaries of the region being revolved

When to use it: Whenever the region under a single curve (with no gap between the curve and the axis of rotation) is revolved around that axis.

V = π ∫ₐᵇ ( [R(x)]² − [r(x)]² ) dx

Rotating a region between two curves around an axis sweeps out a solid with a hole through it; each cross-section is a "washer" — a disk of outer radius R(x) with a smaller disk of inner radius r(x) removed.

R(x)

— the outer radius — distance from the axis to the farther boundary curve

r(x)

— the inner radius — distance from the axis to the nearer boundary curve

When to use it: Whenever the region between two curves (with a gap between the region and the axis of rotation) is revolved around that axis.

Total reading = MSR + (VC × LC), where LC = 1 MSD − 1 VSD

A vernier caliper reads a length more precisely than a plain ruler by adding a small correction from its sliding vernier scale to the main scale reading.

MSR

— main scale reading — the whole-division reading on the fixed main scale, in millimetres (mm)

VC

— vernier coincidence — the number of the vernier division that lines up exactly with a main scale division

LC

— least count — the smallest length the instrument can measure, in millimetres (mm)

MSD, VSD

— the size of one main scale division and one vernier scale division, in millimetres (mm)

When to use it: Whenever measuring a length, diameter, or depth more precisely than a plain ruler allows, using a vernier caliper.

Total reading = MSR + (CSR × LC), where LC = pitch / number of circular scale divisions

A micrometer screw gauge measures very small lengths (like wire diameters) by converting a full turn of a fine screw into a tiny, precisely known linear distance.

MSR

— main scale reading — the reading on the linear scale on the sleeve, in millimetres (mm)

CSR

— circular scale reading — the division on the rotating thimble that lines up with the main scale's reference line

LC

— least count — the smallest length the instrument can measure, in millimetres (mm)

pitch

— the linear distance the spindle advances in one full turn of the thimble, in millimetres (mm)

When to use it: Whenever measuring a very small length — such as the diameter of a wire or the thickness of a sheet — more precisely than a vernier caliper allows.

T = 2π√(L / g)

A simple pendulum swings back and forth with a period that depends only on its length and the local gravitational acceleration — not on its mass or how wide it swings (for small angles). This steady, repeatable period is exactly what makes a pendulum useful as a time standard.

T

— the period — the time for one complete swing (there and back), in seconds (s)

L

— the length of the pendulum, from the pivot to the centre of the bob, in metres (m)

g

— the acceleration due to gravity, approximately 9.8 m/s²

When to use it: Whenever finding the period of a swinging pendulum, or using a pendulum's period to measure time or find g.

speed = distance / time

How fast an object is moving, regardless of direction.

speed

— rate of motion, in metres per second (m/s)

distance

— total path length travelled, in metres (m)

time

— time taken, in seconds (s)

When to use it: Whenever you need how fast something moved, without needing to know its direction.

velocity = displacement / time

Speed in a specific direction.

velocity

— rate of displacement, in metres per second (m/s), in a stated direction

displacement

— straight-line distance from start to finish, in a direction, in metres (m)

time

— time taken, in seconds (s)

When to use it: Whenever direction matters, not just how fast something is moving.

a = (v − u) / t

How quickly velocity is changing.

a

— acceleration, in metres per second squared (m/s²)

v

— final velocity, in m/s

u

— initial velocity, in m/s

t

— time taken for the change, in seconds (s)

When to use it: Whenever an object is speeding up, slowing down, or changing direction, and you need to measure the rate.

v = u + at

Final velocity from initial velocity, acceleration, and time.

v

— final velocity (m/s)

u

— initial velocity (m/s)

a

— acceleration (m/s²)

t

— time (s)

When to use it: Whenever you know initial velocity, acceleration and time, and need the final velocity.

s = ut + ½at²

Displacement from initial velocity, acceleration, and time.

s

— displacement (m)

u

— initial velocity (m/s)

a

— acceleration (m/s²)

t

— time (s)

When to use it: Whenever you know initial velocity, acceleration and time, and need the distance travelled.

v² = u² + 2as

Final velocity from initial velocity, acceleration, and displacement — no time needed.

v

— final velocity (m/s)

u

— initial velocity (m/s)

a

— acceleration (m/s²)

s

— displacement (m)

When to use it: Whenever time is unknown but displacement is, and you need velocity or acceleration.

F = ma

The net (resultant) force acting on an object equals its mass times its acceleration. A bigger force produces a bigger acceleration; a bigger mass resists acceleration more, for the same force.

F

— net force acting on the object, in newtons (N)

m

— mass of the object, in kilograms (kg)

a

— acceleration produced, in metres per second squared (m/s²)

When to use it: Whenever a net force and a mass are known and the resulting acceleration is needed (or any one of the three, given the other two).

F = G m₁m₂ / r²

Every pair of masses in the universe attracts each other with a force that grows with both masses, and shrinks rapidly (as the inverse square) with the distance between their centres.

F

— gravitational force between the two masses, in newtons (N)

G

— the universal gravitational constant, 6.67 × 10⁻¹¹ N·m²/kg²

m₁, m₂

— the two masses, in kilograms (kg)

r

— the distance between the centres of the two masses, in metres (m)

When to use it: Whenever the gravitational force of attraction between two known masses at a known separation needs to be found.

F(friction) = μN

The maximum friction force between two surfaces is proportional to how hard the surfaces are pressed together (the normal reaction force); the constant of proportionality depends only on the two materials in contact.

F(friction)

— the friction force, in newtons (N)

μ

— the coefficient of friction between the two surfaces (dimensionless — no units)

N

— the normal reaction force pressing the surfaces together, in newtons (N)

When to use it: Whenever the maximum friction force between two surfaces needs to be found, given the coefficient of friction and the normal force.

W = mg

Weight is the force of gravity pulling on an object's mass. Unlike mass, weight depends on the local gravitational field strength, so the same object weighs less on the Moon than on Earth.

W

— weight, in newtons (N)

m

— mass of the object, in kilograms (kg)

g

— the local gravitational field strength (acceleration due to gravity), in N/kg or m/s² — approximately 9.8 on Earth's surface

When to use it: Whenever an object's weight needs to be found from its mass, at a known location.

p = mv; impulse J = FΔt = Δp

Momentum measures how hard it is to stop a moving object — it combines mass and velocity. Impulse is the change in momentum a force produces, and equals that force multiplied by the time it acts for.

p

— momentum, in kilogram-metres per second (kg·m/s)

m

— mass, in kilograms (kg)

v

— velocity, in metres per second (m/s)

J

— impulse, in newton-seconds (N·s), numerically equal to kg·m/s

F

— force applied, in newtons (N)

Δt

— the time interval the force acts for, in seconds (s)

Δp

— the resulting change in momentum, in kg·m/s

When to use it: Whenever finding an object's momentum, or the change in momentum (impulse) produced by a force acting for a known time.

P = F / A

Pressure measures how concentrated a force is — the same force spread over a smaller area produces a much bigger pressure.

P

— pressure, in pascals (Pa), where 1 Pa = 1 N/m²

F

— force applied perpendicular to the surface, in newtons (N)

A

— area over which the force acts, in square metres (m²)

When to use it: Whenever the pressure produced by a force acting over a known area needs to be found (or either of the other two quantities, given the other two).

P = hρg

The pressure at a depth within a liquid depends only on how deep the point is, the density of the liquid, and gravity — not on the shape or width of the container.

P

— pressure due to the liquid column, in pascals (Pa)

h

— depth below the liquid surface, in metres (m)

ρ

— density of the liquid, in kilograms per cubic metre (kg/m³)

g

— the acceleration due to gravity, approximately 9.8 m/s²

When to use it: Whenever the pressure at a given depth within a liquid needs to be found — for example, water pressure at the bottom of a tank, or atmospheric pressure modeled as a column of air.

ρ = m / V

Density measures how tightly packed the mass of a substance is — how much mass is squeezed into each unit of volume.

ρ

— density, in kilograms per cubic metre (kg/m³), or grams per cubic centimetre (g/cm³)

m

— mass of the substance, in kilograms (kg) or grams (g)

V

— volume of the substance, in cubic metres (m³) or cubic centimetres (cm³)

When to use it: Whenever a substance's density needs to be found from its mass and volume (or either of the other two, given the other two).

RD = ρ(substance) / ρ(water)

Relative density compares a substance's density directly against water's — a relative density greater than 1 means the substance is denser than water (and sinks in it); less than 1 means it is less dense (and floats).

RD

— relative density (specific gravity) — a pure ratio, with no units

ρ(substance)

— density of the substance, in kg/m³ or g/cm³

ρ(water)

— density of water, 1000 kg/m³ (or 1 g/cm³), in the same units as ρ(substance)

When to use it: Whenever comparing a substance's density to water's, or predicting whether it will float or sink in water.

W = Fd

Work is done whenever a force moves an object through a distance in the direction of the force. No work is done if the object does not move, or if the force has no component in the direction of motion.

W

— work done, in joules (J)

F

— force applied, in newtons (N)

d

— distance moved in the direction of the force, in metres (m)

When to use it: Whenever the work done by a force moving an object through a known distance needs to be found.

P = W / t

Power measures how quickly work is done, or how quickly energy is transferred — the same amount of work done faster means a greater power.

P

— power, in watts (W), where 1 W = 1 J/s

W

— work done (or energy transferred), in joules (J)

t

— time taken, in seconds (s)

When to use it: Whenever the rate at which work is done, or energy is transferred, needs to be found.

KE = ½mv²

Kinetic energy is the energy an object has because it is moving. It grows with the SQUARE of speed, so doubling the speed of an object quadruples its kinetic energy.

KE

— kinetic energy, in joules (J)

m

— mass of the object, in kilograms (kg)

v

— speed of the object, in metres per second (m/s)

When to use it: Whenever the energy of a moving object needs to be found from its mass and speed.

GPE = mgh

Gravitational potential energy is the energy an object has because of its height above a reference level — it is the energy that would be released if the object fell back down to that level.

GPE

— gravitational potential energy, in joules (J)

m

— mass of the object, in kilograms (kg)

g

— the acceleration due to gravity, approximately 9.8 m/s²

h

— height above the reference level, in metres (m)

When to use it: Whenever the gravitational potential energy of an object at a known height needs to be found.

EPE = ½kx²

Elastic potential energy is the energy stored in a stretched or compressed elastic object, such as a spring — it grows with the square of the extension, so stretching a spring twice as far stores four times the energy.

EPE

— elastic potential energy, in joules (J)

k

— the spring constant — a measure of the spring's stiffness, in newtons per metre (N/m)

x

— the extension or compression from the natural length, in metres (m)

When to use it: Whenever the energy stored in a stretched or compressed spring needs to be found.

Q = mcΔT

The heat energy needed to change a substance's temperature depends on its mass, its specific heat capacity (how much energy it takes to heat that particular material), and the size of the temperature change.

Q

— heat energy supplied or removed, in joules (J)

m

— mass of the substance, in kilograms (kg)

c

— specific heat capacity of the substance, in joules per kilogram per kelvin (J/(kg·K))

ΔT

— the change in temperature, in kelvin (K) or degrees Celsius (°C) — the SIZE of the change is the same in both scales

When to use it: Whenever the heat energy needed to change the temperature of a known mass of substance by a known amount needs to be found.

T(K) = T(°C) + 273; T(°F) = (9/5)T(°C) + 32

The Kelvin, Celsius, and Fahrenheit scales all measure the same physical temperature, just with different zero points and different-sized degrees — these formulas convert a reading on one scale to another.

T(K)

— temperature in kelvin (K) — the SI unit, starting at absolute zero

T(°C)

— temperature in degrees Celsius (°C)

T(°F)

— temperature in degrees Fahrenheit (°F)

When to use it: Whenever a temperature needs to be converted between the Celsius, Kelvin, or Fahrenheit scales.

ΔL = L₀αΔT

When a solid is heated, it expands — the amount it expands is proportional to its original length, how much its temperature changes, and a material-specific property called the coefficient of linear expansion.

ΔL

— the change in length, in metres (m)

L₀

— the original length, in metres (m)

α

— the coefficient of linear expansion of the material, in per kelvin (K⁻¹)

ΔT

— the change in temperature, in kelvin (K) or degrees Celsius (°C)

When to use it: Whenever the change in length of a solid due to a temperature change needs to be found.

v = fλ

The speed a wave travels at equals how many complete waves pass a point each second (frequency) multiplied by the length of each wave (wavelength).

v

— wave speed, in metres per second (m/s)

f

— frequency, in hertz (Hz)

λ

— wavelength, in metres (m)

When to use it: Whenever the speed, frequency, or wavelength of a wave needs to be found, given the other two.

T = 1 / f

The period (the time for one complete wave to pass) and the frequency (how many complete waves pass per second) are reciprocals of each other.

T

— period, in seconds (s)

f

— frequency, in hertz (Hz)

When to use it: Whenever converting between the period and frequency of a wave.

d = vt / 2

A sound wave sent toward a distant surface travels there AND back before the echo is heard, so the total time measured corresponds to twice the distance — this formula divides that out.

d

— distance to the reflecting surface, in metres (m)

v

— speed of sound in the medium, in metres per second (m/s)

t

— total time between the sound being sent and the echo being heard, in seconds (s)

When to use it: Whenever the distance to a reflecting surface (a cliff, the sea floor, an obstacle) needs to be found from the round-trip time of an echo.

1/f = 1/v + 1/u; magnification m = −v/u = h(i)/h(o)

The mirror equation relates the distances of the object, the image, and the focal point for a curved mirror. Magnification compares the size of the image to the size of the object, and also reveals whether the image is upright or inverted.

f

— the mirror's focal length, in metres or centimetres (negative for a concave mirror, positive for a convex mirror, using the sign convention where the object side is negative)

u

— object distance from the mirror (always negative, since the object is always in front of the mirror)

v

— image distance from the mirror (negative for a real image in front of the mirror, positive for a virtual image behind it)

m

— magnification (dimensionless) — a negative value means an inverted image, positive means upright; |m| > 1 means enlarged, |m| < 1 means diminished

h(i), h(o)

— image height and object height, in the same units

When to use it: Whenever the image distance, object distance, focal length, or magnification for a curved mirror needs to be found, given the other relevant quantities.

F = k q₁q₂ / r²

The electric force between two charges grows with the size of both charges, and shrinks rapidly (as the inverse square) with the distance between them — mirroring the same inverse-square pattern seen in gravity, but very much stronger.

F

— the electric force between the two charges, in newtons (N) — attractive for opposite charges, repulsive for like charges

k

— Coulomb's constant, about 9 × 10⁹ N·m²/C²

q₁, q₂

— the two charges, in coulombs (C)

r

— the distance between the two charges, in metres (m)

When to use it: Whenever the electric force between two known point charges at a known separation needs to be found.

Q = ne

Electric charge always comes in whole-number multiples of a single, smallest possible unit — the charge on one electron (or proton). Charge cannot be divided into an arbitrarily small fraction.

Q

— total charge, in coulombs (C)

n

— the number of electrons (or protons) — a whole number

e

— the elementary charge, 1.6 × 10⁻¹⁹ C — the smallest unit of charge

When to use it: Whenever the total charge on an object needs to be found from a number of excess (or missing) electrons, or vice versa.

E = hf = hc/λ

Light delivers its energy in discrete packets called photons. The energy of a single photon depends only on the frequency (or, equivalently, the wavelength) of the light — higher-frequency light delivers more energetic photons.

E

— energy of one photon, in joules (J)

h

— Planck's constant, 6.63 × 10⁻³⁴ J·s

f

— frequency of the light, in hertz (Hz)

c

— the speed of light, 3 × 10⁸ m/s

λ

— wavelength of the light, in metres (m)

When to use it: Whenever the energy carried by a single photon of light needs to be found from its frequency or wavelength.

vₓ = v cosθ, vy = v sinθ

Any vector at an angle can be split into a horizontal (x) component and a vertical (y) component that together produce the same effect as the original vector.

v

— Magnitude of the original vector (e.g. speed, m/s)

θ

— Angle the vector makes with the horizontal (x) axis, in degrees or radians

vₓ

— Horizontal (x) component of the vector

vy

— Vertical (y) component of the vector

When to use it: Use whenever a vector (velocity, force, displacement) acts at an angle and you need to analyse its horizontal and vertical effects separately.

R = √(vₓ² + vy²), θ = tan⁻¹(vy / vₓ)

Given two perpendicular components of a vector, this formula recombines them into a single resultant vector with a magnitude and direction.

R

— Magnitude of the resultant vector

vₓ

— Horizontal (x) component

vy

— Vertical (y) component

θ

— Angle of the resultant above the x-axis

When to use it: Use to combine two perpendicular components (e.g. two forces, or a velocity's x and y parts) back into one overall vector.

T = 2u sinθ / g

The total time a projectile stays in the air, from launch until it returns to its launch height.

T

— Time of flight, in seconds

u

— Initial launch speed, in m/s

θ

— Launch angle above the horizontal

g

— Acceleration due to gravity, 9.8 m/s²

When to use it: Use to find how long a projectile launched at an angle stays airborne before landing at the same height it was launched from.

H = u² sin²θ / 2g

The highest vertical point a projectile reaches, where its vertical velocity momentarily becomes zero.

H

— Maximum height, in metres

u

— Initial launch speed, in m/s

θ

— Launch angle above the horizontal

g

— Acceleration due to gravity, 9.8 m/s²

When to use it: Use to find the peak height of a projectile's trajectory.

R = u² sin2θ / g

The total horizontal distance a projectile travels before landing back at its launch height.

R

— Horizontal range, in metres

u

— Initial launch speed, in m/s

θ

— Launch angle above the horizontal

g

— Acceleration due to gravity, 9.8 m/s²

When to use it: Use to find how far a projectile travels horizontally — note the range is maximised when θ = 45°, since sin2θ is then at its maximum value of 1.

T = 2πr / v, f = 1/T

Relates how long one full revolution takes (period) to the radius and speed of circular motion, and how frequency is the reciprocal of period.

T

— Period — time for one complete revolution, in seconds

r

— Radius of the circular path, in metres

v

— Constant speed along the circular path, in m/s

f

— Frequency — number of revolutions per second, in Hz

When to use it: Use to relate the speed, radius, period, and frequency of any object moving at constant speed in a circle.

a = v² / r

The acceleration directed toward the centre of a circular path, responsible for continuously changing the direction of an object's velocity.

a

— Centripetal acceleration, in m/s²

v

— Speed along the circular path, in m/s

r

— Radius of the circular path, in metres

When to use it: Use whenever an object moves at constant speed along a circular path and you need the acceleration causing that curved motion.

F = mv² / r

The net force directed toward the centre of a circular path that is required to keep an object moving in that circle.

F

— Centripetal force, in newtons (N)

m

— Mass of the object, in kg

v

— Speed along the circular path, in m/s

r

— Radius of the circular path, in metres

When to use it: Use to find the net inward force needed to keep an object of known mass moving in a circle of known radius and speed.

A · B = AB cosθ

The scalar product of two vectors gives a single number (scalar), representing how much one vector points in the direction of the other.

A · B

— The scalar (dot) product, a number with no direction

A, B

— Magnitudes of the two vectors

θ

— The angle between the two vectors

When to use it: Use when combining two vectors should produce a plain number, such as calculating work done (force · displacement).

|A × B| = AB sinθ

The vector product of two vectors gives a new vector, perpendicular to both original vectors, with a magnitude that depends on how perpendicular the two original vectors are to each other.

A × B

— The vector (cross) product, itself a vector

A, B

— Magnitudes of the two vectors

θ

— The angle between the two vectors

When to use it: Use when combining two vectors should produce a new vector perpendicular to both, such as calculating torque (r × F) or angular momentum.

τ = r F sinθ

Torque measures the turning effect of a force applied at some distance from a pivot — it depends on the force, the distance from the pivot, and the angle between them.

τ

— Torque, in newton-metres (N·m)

r

— Distance from the pivot (axis of rotation) to where the force is applied, in metres

F

— Magnitude of the applied force, in newtons

θ

— Angle between the force direction and the line from the pivot to the point of application

When to use it: Use to calculate the turning effect of a force about a pivot point — maximum torque occurs when the force is applied perpendicular to r (θ = 90°).

I = mr²

Moment of inertia measures an object's resistance to changes in its rotational motion — for a point mass, it depends on the mass and how far that mass is from the axis of rotation.

I

— Moment of inertia, in kg·m²

m

— Mass of the point/particle, in kg

r

— Distance from the axis of rotation, in metres

When to use it: Use for a point mass or as the building block for finding the total moment of inertia of an extended object (by summing mr² for every mass element).

τ = Iα

This is the rotational equivalent of Newton's second law (F = ma) — net torque produces angular acceleration, with moment of inertia playing the role mass plays in linear motion.

τ

— Net torque, in N·m

I

— Moment of inertia, in kg·m²

α

— Angular acceleration, in rad/s²

When to use it: Use to find the angular acceleration produced by a net torque on a rotating object, or to find the torque needed to produce a given angular acceleration.

ΣF = 0 and Στ = 0

An object is in complete (static) equilibrium only when both the net force AND the net torque acting on it are zero — the first condition prevents linear acceleration, the second prevents rotational (angular) acceleration.

ΣF

— The vector sum of all forces acting on the object

Στ

— The sum of all torques acting on the object, about any chosen pivot

When to use it: Use to analyse any object that is not accelerating linearly or rotationally — e.g. a beam resting on supports, or a ladder leaning against a wall.

L = Iω

Angular momentum measures the "quantity of rotational motion" an object has — the rotational equivalent of linear momentum (p = mv).

L

— Angular momentum, in kg·m²/s

I

— Moment of inertia, in kg·m²

ω

— Angular velocity, in rad/s

When to use it: Use to calculate the angular momentum of a rotating object, and as the basis for applying conservation of angular momentum.

I₁ω₁ = I₂ω₂

When no external torque acts on a system, its total angular momentum stays constant — so if its moment of inertia changes, its angular velocity must change to compensate.

I₁, ω₁

— Moment of inertia and angular velocity at an initial time

I₂, ω₂

— Moment of inertia and angular velocity at a later time

When to use it: Use whenever a rotating system changes shape (redistributing its mass) with no external torque acting on it, such as a spinning figure skater pulling in their arms.

P = F / A

Pressure is the amount of force acting perpendicular to a surface, spread over the area of that surface.

P

— Pressure, in pascals (Pa), where 1 Pa = 1 N/m²

F

— Force acting perpendicular to the surface, in newtons

A

— Area over which the force is spread, in m²

When to use it: Use whenever you need to relate a force to the area it acts over — the same force spread over a smaller area produces much greater pressure.

P = hρg

The pressure exerted by a column of liquid depends on the height of the column, the density of the liquid, and gravity — a barometer uses this to measure atmospheric pressure from the height of a mercury column it supports.

P

— Pressure at the base of the column, in Pa

h

— Height of the liquid column, in metres

ρ

— Density of the liquid (e.g. mercury, 13,600 kg/m³), in kg/m³

g

— Acceleration due to gravity, 9.8 m/s²

When to use it: Use to find atmospheric pressure from the height of mercury a barometer supports, or the pressure exerted by any column of liquid of known height.

P = P₀ + hρg

The total pressure at a depth in a liquid equals the pressure already present at the surface (usually atmospheric pressure) plus the additional pressure from the weight of the liquid above that depth.

P

— Total pressure at depth h, in Pa

P₀

— Pressure at the liquid surface (often atmospheric pressure), in Pa

h

— Depth below the surface, in metres

ρ

— Density of the liquid, in kg/m³

g

— Acceleration due to gravity, 9.8 m/s²

When to use it: Use to find the total pressure at any depth within a liquid, such as at the bottom of a swimming pool or tank.

ΔP = hρg

A manometer measures the pressure of a gas by comparing it to atmospheric pressure, using the height difference between two liquid columns connected to the gas and the open air.

ΔP

— The pressure difference between the gas and the atmosphere, in Pa

h

— The height difference between the two liquid columns in the manometer, in metres

ρ

— Density of the manometer liquid, in kg/m³

g

— Acceleration due to gravity, 9.8 m/s²

When to use it: Use to find a gas's pressure relative to atmospheric pressure from the height difference observed in a manometer's liquid columns.

Fᵦ = ρVg

The upward buoyant force on an object submerged (fully or partly) in a fluid equals the weight of the fluid that the object displaces.

Fᵦ

— Buoyant force, in newtons

ρ

— Density of the fluid the object is submerged in, in kg/m³

V

— Volume of fluid displaced by the object, in m³

g

— Acceleration due to gravity, 9.8 m/s²

When to use it: Use to find the upward force a fluid exerts on a submerged or floating object, and to determine whether an object will float or sink.

F₁/A₁ = F₂/A₂

Pressure applied to an enclosed fluid is transmitted equally throughout the fluid — a small force on a small piston can therefore produce a much larger force on a larger piston, since the pressure (F/A) must be the same at both.

F₁, A₁

— Force applied to, and area of, the smaller (input) piston

F₂, A₂

— Resulting force on, and area of, the larger (output) piston

When to use it: Use to calculate the force multiplication produced by a hydraulic system, such as a car jack or hydraulic press.

P = W / t

Power is the rate at which work is done or energy is transferred — how quickly energy is converted from one form to another.

P

— Power, in watts (W), where 1 W = 1 J/s

W

— Work done (or energy transferred), in joules

t

— Time taken, in seconds

When to use it: Use whenever you need to compare how quickly different amounts of work are done, or find the time needed to do a certain amount of work at a known power.

η = (useful energy output / total energy input) × 100%

Efficiency measures what fraction of the energy put into a machine or process is converted into useful output, rather than being wasted (usually as heat).

η

— Efficiency, expressed as a percentage

useful energy output

— The energy converted into the intended, useful form, in joules

total energy input

— The total energy supplied to the machine or process, in joules

When to use it: Use to evaluate how well a machine or energy-conversion process performs — no real machine achieves 100% efficiency, since some energy is always lost, usually to heat or friction.

F = kx

Within its elastic limit, the force needed to stretch or compress a spring is directly proportional to how far it is stretched or compressed.

F

— Applied force (or restoring force), in newtons

k

— Spring constant — a measure of the spring's stiffness, in N/m

x

— Extension or compression from the natural (unstretched) length, in metres

When to use it: Use to find the force needed to stretch a spring by a known amount, or the extension produced by a known force — valid only within the spring's elastic limit.

E = ½kx²

The energy stored in a stretched or compressed spring, which can be recovered as the spring returns to its natural length.

E

— Elastic potential energy, in joules

k

— Spring constant, in N/m

x

— Extension or compression from the natural length, in metres

When to use it: Use to find the energy stored in a stretched or compressed spring, e.g. to analyse energy transfers in systems using springs.

1 cal = 4.186 J

The calorie (an older heat unit, defined as the heat needed to raise 1 g of water by 1°C) converts to the SI unit of energy, the joule, using this fixed conversion factor.

cal

— Calories — a traditional unit of heat energy

J

— Joules — the SI unit of energy

When to use it: Use to convert between calories and joules when heat energy is given in either unit.

C = Q / ΔT

Thermal capacity is the amount of heat energy needed to raise the temperature of an entire object (of any mass) by one degree.

C

— Thermal capacity, in J/°C (or J/K)

Q

— Heat energy supplied, in joules

ΔT

— Temperature change produced, in °C (or K)

When to use it: Use to describe how much heat a specific object (not a material in general) needs to change temperature — thermal capacity depends on the object's total mass and material.

Q = mcΔT

Specific heat capacity describes how much heat energy is needed to raise the temperature of one kilogram of a specific MATERIAL by one degree — a property of the material itself, independent of how much of it you have.

Q

— Heat energy supplied (or released), in joules

m

— Mass of the substance, in kg

c

— Specific heat capacity of the material, in J/(kg·°C)

ΔT

— Temperature change, in °C (or K)

When to use it: Use to calculate the heat energy needed to change the temperature of a known mass of a specific material by a known amount.

Heat lost by hot object = Heat gained by cold object

When a hot object and a cold object are in thermal contact (with no heat lost to the surroundings), heat flows from the hot object to the cold one until they reach a common final temperature — the heat lost by one exactly equals the heat gained by the other.

m₁c₁(T₁ − Tf)

— Heat lost by the hotter object (mass m₁, specific heat c₁, initial temperature T₁), cooling to the final temperature Tf

m₂c₂(Tf − T₂)

— Heat gained by the cooler object (mass m₂, specific heat c₂, initial temperature T₂), warming to the final temperature Tf

When to use it: Use to find an unknown final temperature, mass, or specific heat capacity when two substances at different temperatures reach thermal equilibrium together.

Q = mLᵥ

The heat energy needed to convert a mass of liquid into vapour (or vice versa) at a constant temperature (the boiling point), with no temperature change during the process itself.

Q

— Heat energy required, in joules

m

— Mass of the substance changing state, in kg

Lᵥ

— Specific latent heat of vaporization of the substance, in J/kg

When to use it: Use to find the heat energy needed to boil (or condense) a known mass of a substance at its boiling point.

Q = mLf

The heat energy needed to convert a mass of solid into liquid (or vice versa) at a constant temperature (the melting point), with no temperature change during the process itself.

Q

— Heat energy required, in joules

m

— Mass of the substance changing state, in kg

Lf

— Specific latent heat of fusion of the substance, in J/kg

When to use it: Use to find the heat energy needed to melt (or freeze) a known mass of a substance at its melting point.

distance between adjacent nodes = λ/2

In a stationary (standing) wave, nodes (points of zero displacement) are evenly spaced, each separated by half a wavelength.

λ

— Wavelength of the wave, in metres

When to use it: Use to find the wavelength of a stationary wave from the measured distance between two adjacent nodes (or antinodes).

f = (1/2L)√(T/μ)

The lowest (fundamental) frequency at which a stretched string vibrates depends on its length, the tension applied, and its mass per unit length.

f

— Fundamental frequency, in Hz

L

— Length of the string, in metres

T

— Tension in the string, in newtons

μ

— Mass per unit length of the string (linear density), in kg/m

When to use it: Use to find the fundamental (lowest) frequency a string produces, or to see how changing tension, length, or thickness affects pitch.

L = λ/4 (first resonance)

A pipe closed at one end (like a resonance tube with water) first resonates when its air column length equals one-quarter of the sound wavelength.

L

— Length of the air column at first resonance, in metres

λ

— Wavelength of the sound wave, in metres

When to use it: Use to find the wavelength (and hence the speed) of sound from the air column length at which a closed resonance tube first resonates with a tuning fork of known frequency.

f = v / 2L

A pipe open at both ends produces its fundamental (lowest) frequency when its length equals half the sound wavelength.

f

— Fundamental frequency, in Hz

v

— Speed of sound in air, in m/s

L

— Length of the pipe, in metres

When to use it: Use to find the fundamental frequency (or required length) of an organ pipe open at both ends.

I = P / A

Intensity measures how concentrated a wave's power is, spreading the total power carried by the wave over the area it passes through.

I

— Intensity, in W/m²

P

— Power carried by the wave, in watts

A

— Area the power is spread over, in m²

When to use it: Use to compare how "strong" a wave is at different distances from its source, or to find the power delivered over a given area.

c = fλ

Like any wave, light's speed equals the product of its frequency and wavelength.

c

— Speed of light (in vacuum, ≈ 3 × 10⁸ m/s)

f

— Frequency of the light, in Hz

λ

— Wavelength of the light, in metres

When to use it: Use to find the wavelength or frequency of light when the other is known, using the known speed of light.

n₁ sinθ₁ = n₂ sinθ₂

When light crosses the boundary between two media, the product of each medium's refractive index and the sine of the angle (measured from the normal) is the same on both sides.

n₁, n₂

— Refractive indices of the first and second media

θ₁

— Angle of incidence, measured from the normal, in the first medium

θ₂

— Angle of refraction, measured from the normal, in the second medium

When to use it: Use to find an unknown angle of incidence or refraction, or an unknown refractive index, when light crosses a boundary between two media.

n = c / v

A medium's refractive index measures how much slower light travels in it compared to in a vacuum.

n

— Refractive index of the medium (no units, always ≥ 1)

c

— Speed of light in a vacuum, ≈ 3 × 10⁸ m/s

v

— Speed of light in the medium, in m/s

When to use it: Use to find a medium's refractive index from the speed of light within it, or to find that speed if the refractive index is known.

sinθc = 1/n

The critical angle is the angle of incidence (in the denser medium) at which the refracted ray travels exactly along the boundary (refraction angle = 90°) — beyond this angle, total internal reflection occurs.

θc

— Critical angle, measured from the normal

n

— Refractive index of the denser medium (light travels from denser to less dense medium)

When to use it: Use to find the critical angle for light travelling from a denser medium (like glass or water) toward a less dense one (like air), given the denser medium's refractive index.

1/v − 1/u = 1/f

Relates an object's distance from a lens, the resulting image distance, and the lens's focal length (using the convention that distances are measured from the lens, with light travelling in the positive direction).

v

— Image distance from the lens

u

— Object distance from the lens (negative, since the object is on the incoming side)

f

— Focal length of the lens (positive for a converging lens, negative for a diverging lens)

When to use it: Use to find the image distance, object distance, or focal length of a lens, given the other two.

m = v/u

Magnification compares the size of the image produced by a lens to the size of the original object, using the ratio of image distance to object distance.

m

— Magnification (no units) — |m| > 1 means the image is larger than the object

v

— Image distance from the lens

u

— Object distance from the lens

When to use it: Use to find how much larger or smaller an image is compared to the original object, once v and u are known.

P = 1/f

A lens's power measures how strongly it converges (or diverges) light — a shorter focal length means a more powerful lens.

P

— Power of the lens, in dioptres (D), where 1 D = 1 m⁻¹

f

— Focal length of the lens, in metres (positive for converging, negative for diverging)

When to use it: Use to find the optical power of a lens (as prescribed for glasses, for instance) from its focal length, or vice versa.

F = kq₁q₂ / r²

The electric force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them.

F

— Electric force between the charges, in newtons (attractive if opposite signs, repulsive if same sign)

k

— Coulomb's constant, ≈ 8.99 × 10⁹ N·m²/C²

q₁, q₂

— Magnitudes of the two point charges, in coulombs (C)

r

— Distance between the charges, in metres

When to use it: Use to find the electric force between two point charges at a known separation.

E = F / q

Electric field intensity measures the force per unit charge that would be experienced by a small positive test charge placed at a point.

E

— Electric field intensity, in N/C (or equivalently V/m)

F

— Force experienced by the test charge, in newtons

q

— Magnitude of the test charge, in coulombs

When to use it: Use to find the electric field strength at a point from the force it would exert on a known test charge, or to find the force on a charge placed in a known field.

V = kQ / r

Electric potential at a point measures the electric potential energy per unit charge that a small positive test charge would have at that point, due to a source charge Q.

V

— Electric potential, in volts (V)

k

— Coulomb's constant, ≈ 8.99 × 10⁹ N·m²/C²

Q

— Magnitude of the source charge, in coulombs

r

— Distance from the source charge, in metres

When to use it: Use to find the electric potential at a point a known distance from a point charge.

V = Ed

For a uniform electric field between two parallel charged plates, the potential difference between them equals the field strength multiplied by the separation between the plates.

V

— Potential difference between the plates, in volts

E

— Electric field strength between the plates, in V/m (uniform)

d

— Distance (separation) between the plates, in metres

When to use it: Use to find the potential difference across parallel plates given the field strength and separation, or to find the field strength given a known applied voltage.

I = Q / t

Electric current is the rate at which electric charge flows past a point in a circuit.

I

— Current, in amperes (A)

Q

— Charge that flows, in coulombs (C)

t

— Time taken, in seconds

When to use it: Use to find the current from the amount of charge flowing in a given time, or vice versa.

V = IR

For many conductors (at constant temperature), the potential difference across them is directly proportional to the current flowing through them, with resistance as the constant of proportionality.

V

— Potential difference (voltage) across the conductor, in volts

I

— Current flowing through the conductor, in amperes

R

— Resistance of the conductor, in ohms (Ω)

When to use it: Use to find any one of voltage, current, or resistance when the other two are known.

R = ρL / A

A conductor's resistance depends on its material (resistivity), its length, and its cross-sectional area — longer and thinner conductors have higher resistance.

R

— Resistance, in ohms (Ω)

ρ

— Resistivity of the material (a property of the material itself), in Ω·m

L

— Length of the conductor, in metres

A

— Cross-sectional area of the conductor, in m²

When to use it: Use to find a conductor's resistance from its material, length, and cross-sectional area, or to design a resistor with a specific target resistance.

Rₛ = R₁ + R₂ + R₃ + ...

When resistors are connected in series (end to end, one path for current), their resistances simply add together to give the total (equivalent) resistance.

Rₛ

— Total (equivalent) resistance of the series combination, in ohms

R₁, R₂, R₃...

— Resistance of each individual resistor, in ohms

When to use it: Use to find the total resistance of resistors connected end-to-end along a single current path.

1/Rₚ = 1/R₁ + 1/R₂ + 1/R₃ + ...

When resistors are connected in parallel (providing multiple separate paths for current), the reciprocal of the total resistance equals the sum of the reciprocals of each individual resistance.

Rₚ

— Total (equivalent) resistance of the parallel combination, in ohms

R₁, R₂, R₃...

— Resistance of each individual resistor, in ohms

When to use it: Use to find the total resistance of resistors connected across the same two points, providing multiple parallel current paths.

β = I_C / I_B

Current gain measures how much a transistor amplifies a small base current into a much larger collector current.

β

— Current gain (no units) — typically ranging from tens to hundreds

I_C

— Collector current, in amperes

I_B

— Base current, in amperes

When to use it: Use to find how much a transistor amplifies a small input (base) current, or to find the required base current for a target collector current.

Eₙ = −13.6 / n² eV

In Bohr's model of the hydrogen atom, electrons can only occupy specific, quantised energy levels, numbered by n = 1, 2, 3, ... — this formula gives the energy of each allowed level.

Eₙ

— Energy of the electron in level n, in electron-volts (eV)

n

— Principal quantum number — a positive integer (1, 2, 3, ...) labelling the energy level

When to use it: Use to find the energy of an electron in a specific level of a hydrogen atom, or to find the energy released/absorbed when an electron jumps between levels.

KEmax = hf − φ

The maximum kinetic energy of an electron ejected from a metal surface by light equals the energy of the absorbed photon, minus the minimum energy (work function) needed to free the electron from the metal.

KEmax

— Maximum kinetic energy of an ejected electron, in joules (or eV)

h

— Planck's constant, ≈ 6.63 × 10⁻³⁴ J·s

f

— Frequency of the incident light, in Hz

φ

— Work function of the metal — the minimum energy needed to free an electron, in joules (or eV)

When to use it: Use to find the maximum kinetic energy of photoelectrons ejected by light of a known frequency, or to find a metal's work function or the light's frequency, given the other quantities.

v = H₀d

Distant galaxies recede from us at a speed proportional to their distance — the farther away a galaxy is, the faster it appears to be moving away, providing evidence the universe is expanding.

v

— Recession speed of the galaxy, in km/s

H₀

— Hubble's constant, ≈ 70 km/s/Mpc (kilometres per second per megaparsec)

d

— Distance to the galaxy, in megaparsecs (Mpc)

When to use it: Use to find a galaxy's recession speed from its distance, or to estimate distance from an observed recession speed (measured via redshift).

θ = s / r

The angle swept out by a rotating object, measured in radians, found from the arc length traveled divided by the radius.

θ

— angular displacement, in radians (rad)

s

— arc length traveled along the circular path, in metres (m)

r

— radius of the circular path, in metres (m)

When to use it: Whenever the arc length and radius of a circular path are known, and the angle swept (in radians) is needed.

ω = Δθ / Δt

The rate at which angular displacement changes over time — how fast something is rotating.

ω

— angular velocity, in radians per second (rad/s)

Δθ

— change in angular displacement, in radians (rad)

Δt

— time interval, in seconds (s)

When to use it: Whenever the rate of rotation needs to be found from the angle swept and the time it took.

α = Δω / Δt

The rate at which angular velocity changes over time — how quickly a rotation speeds up or slows down.

α

— angular acceleration, in radians per second squared (rad/s²)

Δω

— change in angular velocity, in radians per second (rad/s)

Δt

— time interval, in seconds (s)

When to use it: Whenever a rotation is speeding up or slowing down, and the rate of that change is needed.

ω = ω₀ + αt; θ = ω₀t + ½αt²; ω² = ω₀² + 2αθ

The rotational counterparts of the straight-line "suvat" equations — they relate angular displacement, initial and final angular velocity, angular acceleration, and time, for constant angular acceleration.

ω₀

— initial angular velocity, in radians per second (rad/s)

ω

— final angular velocity, in radians per second (rad/s)

α

— constant angular acceleration, in radians per second squared (rad/s²)

t

— time elapsed, in seconds (s)

θ

— angular displacement over that time, in radians (rad)

When to use it: Whenever a rotation has constant angular acceleration, and any three of the five quantities (ω₀, ω, α, t, θ) are known — pick the equation missing the one unknown that is not needed.

v = rω

The linear (tangential) speed of a point on a rotating object equals its distance from the axis times the angular velocity.

v

— linear (tangential) speed, in metres per second (m/s)

r

— distance from the axis of rotation, in metres (m)

ω

— angular velocity, in radians per second (rad/s)

When to use it: Whenever converting between how fast something spins (angular velocity) and how fast a specific point on it moves through space (linear speed).

aₜ = rα

The linear acceleration of a point on a rotating object, directed along its direction of travel (tangent to the circle) — caused by the rotation speeding up or slowing down.

aₜ

— tangential acceleration, in metres per second squared (m/s²)

r

— distance from the axis of rotation, in metres (m)

α

— angular acceleration, in radians per second squared (rad/s²)

When to use it: Whenever a rotating object's angular acceleration needs to be converted into the linear acceleration of a specific point along its direction of motion.

a_c = v² / r = ω²r

The acceleration of an object moving on a circular path, always directed toward the centre of the circle — it changes the direction of motion, not the speed.

a_c

— centripetal acceleration, in metres per second squared (m/s²)

v

— linear (tangential) speed, in metres per second (m/s)

r

— radius of the circular path, in metres (m)

ω

— angular velocity, in radians per second (rad/s)

When to use it: Whenever an object moves on a circular path at constant speed and the acceleration responsible for curving that path is needed.

T = 1 / f; ω = 2πf = 2π / T

Connects how long one full revolution takes (period), how many revolutions happen per second (frequency), and the angular velocity of the rotation.

T

— period — time for one full revolution, in seconds (s)

f

— frequency — revolutions per second, in hertz (Hz)

ω

— angular velocity, in radians per second (rad/s)

When to use it: Whenever converting between how often something rotates (frequency), how long one rotation takes (period), and its angular velocity.

Fc = mv² / r

The net force needed to keep a mass moving on a circular path of radius r at speed v.

Fc

— centripetal force, in newtons (N)

m

— mass of the object, in kilograms (kg)

v

— linear (tangential) speed, in metres per second (m/s)

r

— radius of the circular path, in metres (m)

When to use it: Whenever you know an object's linear speed and the radius it curves around, and need the net inward force.

Fc = mrω²

The same centripetal force, written using angular velocity instead of linear speed — useful whenever rotation rate (rad/s) is what you are given.

Fc

— centripetal force, in newtons (N)

m

— mass of the object, in kilograms (kg)

r

— radius of the circular path, in metres (m)

ω

— angular velocity, in radians per second (rad/s)

When to use it: Whenever you know how fast something is rotating in radians per second (rather than its linear speed) — substitute v = rω into Fc = mv²/r to get this form.

Re = ρvd / η

A dimensionless number that predicts whether a fluid's flow will be smooth (laminar) or chaotic (turbulent) — comparing the fluid's inertia to its internal friction (viscosity).

Re

— Reynolds number (dimensionless — no units)

ρ

— fluid density, in kilograms per cubic metre (kg/m³)

v

— flow speed, in metres per second (m/s)

d

— characteristic diameter of the flow (e.g. pipe diameter), in metres (m)

η

— dynamic viscosity of the fluid, in pascal-seconds (Pa·s)

When to use it: Whenever predicting or classifying whether a flow is laminar (Re below about 2000) or turbulent (Re above about 4000).

A₁v₁ = A₂v₂

For an incompressible fluid flowing through a pipe, the same volume of fluid must pass every cross-section every second — so where the pipe narrows, the fluid must speed up.

A₁, A₂

— cross-sectional area at two points along the pipe, in square metres (m²)

v₁, v₂

— flow speed at those same two points, in metres per second (m/s)

When to use it: Whenever fluid flows through a pipe or channel that changes cross-sectional area, and the speed at one point needs to be found from the speed and areas elsewhere.

P + ½ρv² + ρgh = constant

Along a streamline in a flowing, incompressible, non-viscous fluid, the sum of pressure energy, kinetic energy per volume, and gravitational potential energy per volume stays constant.

P

— pressure at a point in the fluid, in pascals (Pa)

ρ

— fluid density, in kilograms per cubic metre (kg/m³)

v

— flow speed at that point, in metres per second (m/s)

h

— height of that point above a reference level, in metres (m)

g

— acceleration due to gravity, 9.8 m/s²

When to use it: Whenever relating pressure, speed, and height at two different points along the same flow.

ΔP = ½ρ(v₁² − v₂²); F_lift = ΔP × A

When fluid moves faster over one surface than another at the same height, Bernoulli's equation predicts a pressure difference between the two sides — pushing toward the faster-moving, lower-pressure side.

ΔP

— pressure difference between the two sides, in pascals (Pa)

v₁, v₂

— flow speeds on the two sides, in metres per second (m/s)

F_lift

— net force from the pressure difference, in newtons (N)

A

— surface area the pressure difference acts over, in square metres (m²)

When to use it: Whenever explaining or calculating the force created by a speed difference across two sides of a surface — such as lift on a wing or the pull of an atomizer.

F = ηA(v/d)

The tangential force needed to drag one layer of fluid over another, at speed v, across a fluid layer of thickness d — proportional to how "thick" (viscous) the fluid is.

F

— tangential (viscous) force, in newtons (N)

η

— dynamic viscosity of the fluid, in pascal-seconds (Pa·s)

A

— area of the moving layer in contact with the fluid, in square metres (m²)

v

— relative speed between the two layers, in metres per second (m/s)

d

— perpendicular distance (thickness) between the two layers, in metres (m)

When to use it: Whenever the internal friction force within a flowing fluid, or between a fluid and a moving surface, needs to be found.

γ = F / L

Surface tension is the force per unit length acting along the surface of a liquid, caused by the liquid's molecules pulling on each other more strongly than they are pulled by the air above.

γ

— surface tension coefficient, in newtons per metre (N/m)

F

— force acting along the liquid surface, in newtons (N)

L

— length of the contact line along which the force acts, in metres (m)

When to use it: Whenever the force needed to stretch, break, or hold together a liquid surface (like a soap film) needs to be found from the liquid's surface tension.

h = 2γcosθ / (ρgr)

How high (or low) a liquid rises inside a narrow tube, due to the balance between surface tension pulling the liquid up the tube walls and gravity pulling it back down.

h

— height the liquid rises (or falls) in the tube, in metres (m)

γ

— surface tension coefficient of the liquid, in newtons per metre (N/m)

θ

— contact angle between the liquid and the tube wall, in degrees

ρ

— density of the liquid, in kilograms per cubic metre (kg/m³)

r

— radius of the tube, in metres (m)

When to use it: Whenever the height a liquid rises or falls inside a thin (capillary) tube needs to be predicted.

D = D₀(1+r)ᵗ

Energy demand tends to grow by a roughly constant percentage each year — the same compound-growth pattern as compound interest — letting future demand be projected from a current value and a growth rate.

D

— projected energy demand after time t, in the same units as D₀ (e.g. megawatts, MW)

D₀

— current (initial) energy demand

r

— annual growth rate, as a decimal (e.g. 5% = 0.05)

t

— number of years into the future

When to use it: Whenever projecting how much energy demand will grow over a number of years at a roughly constant annual growth rate.

E = mq

The total energy released by completely burning a fuel equals its mass times its calorific value — the amount of energy stored per kilogram of that particular fuel.

E

— total energy released, in joules (J) or megajoules (MJ)

m

— mass of fuel burned, in kilograms (kg)

q

— the fuel's calorific value (specific energy content), in joules per kilogram (J/kg) or megajoules per kilogram (MJ/kg)

When to use it: Whenever the total energy released by burning a known mass of a non-renewable fuel needs to be found.

P = ½ρAv³

The power carried by moving air through a given area depends on the air density, the swept area of the turbine blades, and — most steeply — the cube of the wind speed.

P

— power available in the wind, in watts (W)

ρ

— density of air, in kilograms per cubic metre (kg/m³) — about 1.2 kg/m³ at sea level

A

— cross-sectional area swept by the turbine's blades, in square metres (m²)

v

— wind speed, in metres per second (m/s)

When to use it: Whenever estimating how much power a wind turbine of a given blade size could generate from a given wind speed.

m_CO2 = m_fuel × EF

The mass of carbon dioxide released by burning a fuel is proportional to the mass of fuel burned, scaled by that fuel's emission factor (how much CO2 each kilogram of fuel releases when burned).

m_CO2

— mass of carbon dioxide released, in kilograms (kg)

m_fuel

— mass of fuel burned, in kilograms (kg)

EF

— emission factor for that fuel, in kilograms of CO2 released per kilogram of fuel burned

When to use it: Whenever estimating the carbon dioxide emissions produced by burning a known mass of a particular fuel.

Q/t = kA(ΔT)/d

The rate at which heat conducts through a material depends on how thermally conductive the material is, the cross-sectional area it flows through, and the temperature difference — but slows down the thicker the material is.

Q/t

— rate of heat transfer, in watts (W)

k

— thermal conductivity of the material, in watts per metre-kelvin (W/(m·K))

A

— cross-sectional area heat flows through, in square metres (m²)

ΔT

— temperature difference across the material, in kelvin (K)

d

— thickness of the material, in metres (m)

When to use it: Whenever the rate of heat conduction through a solid material (like a wall, a metal rod, or a window) needs to be found.

P = εσAT⁴

Every object radiates thermal energy at a rate that depends steeply on its absolute temperature (to the fourth power), scaled by its surface area and how effectively it radiates (its emissivity).

P

— power radiated, in watts (W)

ε

— emissivity of the surface (0 to 1, dimensionless) — 1 for a perfect radiator ('black body')

σ

— the Stefan-Boltzmann constant, 5.67×10⁻⁸ W/(m²·K⁴)

A

— surface area of the radiating object, in square metres (m²)

T

— absolute temperature of the object, in kelvin (K)

When to use it: Whenever the power an object radiates as heat (via electromagnetic radiation) needs to be found from its temperature and surface area.

W = PΔV

When a gas expands or is compressed at constant pressure, the work it does (or has done on it) equals the pressure times the change in its volume — exactly the area under a horizontal line on a pressure-volume graph.

W

— work done by the gas, in joules (J)

P

— the (constant) pressure of the gas, in pascals (Pa)

ΔV

— change in volume (final minus initial), in cubic metres (m³)

When to use it: Whenever the work done by (or on) a gas expanding or compressing at constant pressure needs to be found.

ΔU = Q − W

A statement of energy conservation for a thermodynamic system: the change in a gas's internal energy equals the heat added to it, minus the work it does on its surroundings.

ΔU

— change in internal energy, in joules (J)

Q

— heat added to the system, in joules (J) — negative if heat leaves the system

W

— work done BY the system on its surroundings, in joules (J) — negative if work is done ON the system

When to use it: Whenever relating heat added to a system, work it does, and the resulting change in its internal energy.

η = W/Qₕ = 1 − Qc/Qₕ

A heat engine's efficiency is the fraction of the heat energy it takes in from a hot reservoir that it actually converts into useful work, rather than expelling as waste heat to a cold reservoir.

η

— efficiency (a fraction from 0 to 1, or a percentage)

W

— useful work output, in joules (J)

Qₕ

— heat absorbed from the hot reservoir, in joules (J)

Qc

— heat expelled to the cold reservoir, in joules (J)

When to use it: Whenever the efficiency of a heat engine needs to be found from the heat it takes in and the heat it expels (or the work it produces).

v = 331 + 0.6T

The speed of sound in air increases with temperature — warmer air molecules move faster and transmit the pressure disturbance of a sound wave more quickly.

v

— speed of sound in air, in metres per second (m/s)

T

— air temperature, in degrees Celsius (°C)

When to use it: Whenever the speed of sound in air at a given temperature needs to be estimated (an approximation valid near everyday temperatures).

d = vt / 2

The distance to a wall or other reflecting surface, found from the total round-trip time for a sound to travel out, reflect, and return — divided by 2 since the sound covers the distance twice.

d

— distance to the reflecting surface, in metres (m)

v

— speed of sound in the medium, in metres per second (m/s)

t

— total round-trip time from the sound being made to the echo being heard, in seconds (s)

When to use it: Whenever the distance to a reflecting surface needs to be found from the time it takes to hear an echo.

d_min = (v × t_min) / 2

The human ear needs a minimum gap of about 0.1 s between two sounds to perceive them as separate — so a reflecting surface must be at least this far away for its echo to be heard as distinct from the original sound, rather than blending into it.

d_min

— minimum distance for a distinguishable echo, in metres (m)

v

— speed of sound in the medium, in metres per second (m/s)

t_min

— the minimum time gap the human ear can distinguish, about 0.1 s

When to use it: Whenever determining how far away a surface must be for a person to perceive a distinct echo, rather than the sound simply seeming prolonged.

f' = f(v ± v₀) / (v ∓ v_s)

The frequency an observer hears shifts higher when the source and observer are approaching each other, and lower when they are moving apart, because relative motion changes how many wave crests reach the observer each second.

f'

— the frequency heard by the observer, in hertz (Hz)

f

— the frequency actually emitted by the source, in hertz (Hz)

v

— speed of sound in the medium, in metres per second (m/s)

v₀

— the observer's speed (+ if approaching the source, − if moving away), in metres per second (m/s)

v_s

— the source's speed (− if approaching the observer, + if moving away), in metres per second (m/s)

When to use it: Whenever the frequency heard by an observer needs to be found when a sound source and/or the observer are moving relative to each other.

v = fλ

A wave's speed equals its frequency times its wavelength — how many complete waves pass a point each second, multiplied by the length of each one.

v

— wave speed, in metres per second (m/s)

f

— frequency, in hertz (Hz)

λ

— wavelength, in metres (m)

When to use it: Whenever relating a sound wave's speed, frequency, and wavelength — given any two, the third can be found.

1/f = 1/v + 1/u

A camera's converging lens forms a sharp, focused image on the sensor (or film) exactly where the lens equation predicts, given the lens's focal length and the object's distance.

f

— focal length of the lens, in metres (m)

v

— image distance — from the lens to the focused image, in metres (m)

u

— object distance — from the lens to the object being photographed, in metres (m)

When to use it: Whenever the distance a camera lens must be from its sensor to form a sharp image of an object at a given distance needs to be found.

P = 1/f

A corrective lens's power (in diopters) is the reciprocal of its focal length — for a myopic (near-sighted) eye, a diverging lens with a focal length equal to the eye's far point brings distant objects into focus.

P

— power of the lens, in diopters (D)

f

— focal length needed, in metres (m) — negative for a diverging (myopia-correcting) lens

When to use it: Whenever the power of a corrective lens needs to be found from an eye's near or far point.

M ≈ (L/f₀)(D/fₑ)

A compound microscope's total magnification comes from two lenses working together: the objective lens creates a magnified real image, which the eyepiece lens then magnifies again as a virtual image.

M

— total magnifying power (dimensionless)

L

— tube length — the distance between the objective and eyepiece lenses, in metres (m)

f₀

— focal length of the objective lens, in metres (m)

D

— the near point distance of the eye, conventionally 0.25 m

fₑ

— focal length of the eyepiece lens, in metres (m)

When to use it: Whenever a compound microscope's total magnifying power needs to be estimated from its lens focal lengths and tube length.

M = f₀/fₑ

A telescope's magnifying power is simply the ratio of its objective lens's focal length to its eyepiece lens's focal length — a longer objective or a shorter eyepiece both increase magnification.

M

— magnifying power (dimensionless)

f₀

— focal length of the objective lens, in metres (m)

fₑ

— focal length of the eyepiece lens, in metres (m)

When to use it: Whenever a telescope's magnifying power needs to be found from its two lenses' focal lengths.

E = hf = hc/λ

Light (including laser light) is carried by photons, each with an energy directly proportional to its frequency (or inversely proportional to its wavelength).

E

— energy of a single photon, in joules (J)

h

— Planck's constant, 6.626×10⁻³⁴ J·s

f

— frequency of the light, in hertz (Hz)

c

— speed of light, 3×10⁸ m/s

λ

— wavelength of the light, in metres (m)

When to use it: Whenever the energy carried by a single photon of light (such as laser light of a known wavelength) needs to be found.

θc = sin⁻¹(1/n)

Light traveling inside an optical fibre's core reflects perfectly off the core-cladding boundary (total internal reflection) whenever it strikes at an angle steeper than this critical angle — trapping the light inside the fibre.

θc

— critical angle, in degrees

n

— refractive index of the fibre core relative to the surrounding medium (dimensionless)

When to use it: Whenever the critical angle for total internal reflection inside an optical fibre needs to be found from its core's refractive index.

d sinθ = mλ

A spectrometer's diffraction grating splits light into its component wavelengths, each appearing at its own precise angle — measuring that angle lets the instrument determine the light's exact wavelength.

d

— spacing between adjacent grating lines, in metres (m)

θ

— angle of the diffracted beam from the straight-through direction, in degrees

m

— order of the diffraction maximum (m=1, 2, 3, …)

λ

— wavelength of the light being measured, in metres (m)

When to use it: Whenever a spectrometer's diffraction grating is used to measure or predict the wavelength of light from the angle at which it diffracts.

E = I/d²

The illuminance (brightness of light falling on a surface) from a point source falls off with the square of the distance — moving twice as far away makes a light source appear four times dimmer.

E

— illuminance, in lux (lx)

I

— luminous intensity of the source, in candela (cd)

d

— distance from the source, in metres (m)

When to use it: Whenever the brightness of light falling on a surface at a known distance from a point light source needs to be found.

Δy = λL/d

In a double-slit interference pattern, the spacing between adjacent bright fringes on the screen depends on the wavelength of light, the distance to the screen, and how far apart the two slits are.

Δy

— spacing between adjacent bright fringes, in metres (m)

λ

— wavelength of the light, in metres (m)

L

— distance from the slits to the screen, in metres (m)

d

— distance between the two slits, in metres (m)

When to use it: Whenever the spacing between bright fringes in a double-slit interference pattern needs to be found — or used to work backward to find the light's wavelength.

a sinθ = mλ

Light passing through a single narrow slit spreads out and produces a pattern of dark bands (minima) at specific angles, determined by the slit's width relative to the light's wavelength.

a

— width of the slit, in metres (m)

θ

— angle from the center to the dark band, in degrees

m

— order of the minimum (m=1, 2, 3, …, not m=0)

λ

— wavelength of the light, in metres (m)

When to use it: Whenever the angle of a dark band in a single-slit diffraction pattern needs to be found (or the slit width worked out from a known dark-band angle).

d sinθ = mλ

A diffraction grating's many closely-spaced slits produce sharp, bright bands at precise angles for each wavelength — the same relationship a spectrometer uses, but here describing the grating's own bright-fringe physics rather than instrument measurement.

d

— spacing between adjacent grating lines, in metres (m)

θ

— angle of the bright band from the straight-through direction, in degrees

m

— order of the bright band (m=0, 1, 2, 3, …)

λ

— wavelength of the light, in metres (m)

When to use it: Whenever the angle of a bright diffraction order from a grating needs to be found (or worked backward to find wavelength or grating spacing).

C = Q/V

A capacitor's capacitance measures how much charge it can store per volt of potential difference applied across it — a bigger capacitance stores more charge for the same voltage.

C

— capacitance, in farads (F)

Q

— magnitude of charge stored on each plate, in coulombs (C)

V

— potential difference across the capacitor, in volts (V)

When to use it: Whenever the capacitance of a capacitor needs to be found from the charge it stores and the voltage across it (or vice versa).

C = ε₀εᵣA/d

A parallel-plate capacitor's capacitance depends on the area of its plates (more area, more charge storage), how far apart they are (closer plates, more capacitance), and the insulating material (dielectric) between them.

C

— capacitance, in farads (F)

ε₀

— permittivity of free space, 8.85×10⁻¹² F/m

εᵣ

— relative permittivity (dielectric constant) of the material between the plates (dimensionless; 1 for vacuum/air)

A

— area of overlap between the two plates, in square metres (m²)

d

— separation between the plates, in metres (m)

When to use it: Whenever a parallel-plate capacitor's capacitance needs to be found (or designed) from its physical dimensions and the material between its plates.

E = ½CV²

A charged capacitor stores energy in the electric field between its plates — this energy grows with capacitance, and with the SQUARE of the voltage across it.

E

— energy stored, in joules (J)

C

— capacitance, in farads (F)

V

— voltage across the capacitor, in volts (V)

When to use it: Whenever the energy stored by a charged capacitor needs to be found from its capacitance and voltage.

1/C_series = 1/C₁ + 1/C₂ + …; C_parallel = C₁ + C₂ + …

Combining capacitors changes the total capacitance in opposite ways depending on the arrangement — series combination always reduces total capacitance below the smallest individual value, while parallel combination always increases it.

C_series

— total (equivalent) capacitance of capacitors in series, in farads (F)

C_parallel

— total (equivalent) capacitance of capacitors in parallel, in farads (F)

C₁, C₂, …

— the individual capacitances being combined, in farads (F)

When to use it: Whenever the total (equivalent) capacitance of several capacitors connected in series or in parallel needs to be found.

V = ε − Ir

A real battery's terminal voltage (what a circuit actually sees) is always slightly less than its full EMF, because some voltage is lost driving current through the battery's own internal resistance.

V

— terminal voltage, in volts (V)

ε

— electromotive force (EMF) of the battery, in volts (V)

I

— current flowing through the circuit, in amperes (A)

r

— the battery's internal resistance, in ohms (Ω)

When to use it: Whenever the actual voltage a circuit experiences (as opposed to the battery's ideal EMF) needs to be found, accounting for internal resistance.

Series: ε_total = ε₁+ε₂+…, r_total = r₁+r₂+…; Parallel (identical cells): ε_total = ε, r_total = r/n

Connecting batteries in series adds up their EMFs (and internal resistances) to give a higher voltage; connecting identical batteries in parallel keeps the same EMF but reduces the effective internal resistance, letting more current be supplied.

ε_total

— total (equivalent) EMF of the combination, in volts (V)

r_total

— total (equivalent) internal resistance, in ohms (Ω)

n

— number of identical cells connected in parallel

When to use it: Whenever the total EMF and internal resistance of several batteries connected in series or parallel needs to be found.

P = VI = I²R = V²/R; E = Pt

Electrical power is the rate at which electrical energy is converted (to heat, light, motion, etc.); the total energy used over a time interval is simply power multiplied by time.

P

— electrical power, in watts (W)

V

— voltage across the component, in volts (V)

I

— current through the component, in amperes (A)

R

— resistance of the component, in ohms (Ω)

E

— total electrical energy used, in joules (J)

t

— time, in seconds (s)

When to use it: Whenever the electrical power delivered to a component, or the total energy used over some time, needs to be found.

H = I²Rt

When current flows through a resistor, electrical energy is converted into heat at a rate that grows with the SQUARE of the current — the total heat generated over a time t is I²Rt.

H

— heat generated, in joules (J)

I

— current through the resistor, in amperes (A)

R

— resistance, in ohms (Ω)

t

— time current flows, in seconds (s)

When to use it: Whenever the heat generated by current flowing through a resistance over a given time needs to be found.

R = V²/P

A heating device (like an electric heater or iron) is designed with a specific resistance so that, at its rated operating voltage, it dissipates exactly its intended power as heat.

R

— resistance of the heating element, in ohms (Ω)

V

— operating voltage, in volts (V)

P

— target (rated) power output, in watts (W)

When to use it: Whenever designing (or checking) a heating element's resistance to deliver a specific power at a known operating voltage.

V_rms = V₀/√2

Since an alternating voltage constantly changes, its "effective" value for delivering power is not its peak — the RMS (root-mean-square) voltage is the steady (DC-equivalent) voltage that would deliver the same average power.

V_rms

— root-mean-square (effective) voltage, in volts (V)

V₀

— peak (maximum) voltage of the AC waveform, in volts (V)

When to use it: Whenever converting between the peak voltage of an AC supply and the "effective" voltage that determines its power delivery (the value usually quoted for mains electricity).

Φ = BAcosθ

Magnetic flux measures the total amount of magnetic field passing through a surface — it depends on the field strength, the area of the surface, and the angle between the field and the surface's normal (perpendicular) direction.

Φ

— magnetic flux, in webers (Wb)

B

— magnetic field strength, in tesla (T)

A

— area of the surface (e.g. a coil loop), in square metres (m²)

θ

— angle between the magnetic field and the normal (perpendicular) to the surface, in degrees

When to use it: Whenever the amount of magnetic field passing through a loop or coil needs to be found — the essential first step before applying Faraday's law.

ε = −N(ΔΦ/Δt)

A changing magnetic flux through a coil induces an EMF, proportional to how fast the flux changes and to the number of turns in the coil — the negative sign (Lenz's law) shows the induced EMF always opposes the change that created it.

ε

— induced EMF, in volts (V)

N

— number of turns (loops) in the coil

ΔΦ

— change in magnetic flux through one turn, in webers (Wb)

Δt

— time over which the flux changes, in seconds (s)

When to use it: Whenever the EMF induced by a changing magnetic flux through a coil needs to be found.

Vₛ/Vₚ = Nₛ/Nₚ

A transformer changes AC voltage between its primary and secondary coils in direct proportion to the ratio of their turns — more turns on one side means proportionally higher voltage on that side.

Vₚ

— voltage across the primary coil, in volts (V)

Vₛ

— voltage across the secondary coil, in volts (V)

Nₚ

— number of turns on the primary coil

Nₛ

— number of turns on the secondary coil

When to use it: Whenever a transformer's output (secondary) voltage needs to be found from its input voltage and the ratio of turns on its two coils.

P_loss = I²R

Transmitting power at a HIGHER voltage requires a proportionally LOWER current for the same delivered power — and since power lost as heat in the transmission lines grows with the square of current, high-voltage transmission dramatically reduces wasted energy.

P_loss

— power lost as heat in the transmission line, in watts (W)

I

— current flowing through the transmission line, in amperes (A)

R

— resistance of the transmission line, in ohms (Ω)

When to use it: Whenever the power wasted as heat in a transmission line needs to be found, or to compare the efficiency of transmitting the same power at different voltages.

I_total = I₁+I₂+I₃+…

Household appliances are wired in parallel (so each gets the full mains voltage independently), meaning the total current drawn from the circuit is simply the sum of each appliance's individual current.

I_total

— total current drawn from the circuit, in amperes (A)

I₁, I₂, I₃, …

— current drawn by each individual appliance, in amperes (A)

When to use it: Whenever checking whether a household circuit's total current draw stays within its fuse or circuit breaker's safe rating.

I = V/R

The current that flows through a person's body during an electric shock depends on the voltage they contact and their body's own resistance — a simple application of Ohm's law, but with genuinely dangerous consequences.

I

— current flowing through the body, in amperes (A)

V

— voltage the body is exposed to, in volts (V)

R

— the body's electrical resistance, in ohms (Ω) — far lower when skin is wet than when dry

When to use it: Whenever estimating the danger of an electric shock from a given voltage and the body's resistance under those conditions.

decimal = sum of (bit_i times 2 raised to the power i)

A binary number represents a value using only 0s and 1s, where each digit (bit) stands for a power of 2 based on its position — converting to decimal means adding up the value of every position holding a 1.

bit_i

— the digit (0 or 1) at position i, counted from the rightmost digit (position 0)

2 raised to the power i

— the place value of that position (1, 2, 4, 8, 16, ...)

When to use it: Whenever a binary number (used internally by digital electronics) needs to be converted into an ordinary decimal number, or vice versa.

AND: Y = A AND B; OR: Y = A OR B; NOT: Y = NOT A

Logic gates are the basic building blocks of digital circuits, each following a fixed rule for combining binary (0 or 1) inputs into a binary output.

A, B

— binary inputs (0 or 1)

Y

— the binary output (0 or 1)

When to use it: Whenever the output of a basic logic gate needs to be found for given binary inputs.

fraction remaining = (1/2) raised to the power (t / T_half)

A radioactive sample decays so that its half-life, T_half, is the time for exactly half of it to decay — after n half-lives have passed, only (1/2) to the power n of the original sample remains.

t

— time elapsed, in the same units as the half-life

T_half

— the half-life of the radioactive substance — the time for half of it to decay

When to use it: Whenever the fraction (or amount) of a radioactive sample remaining after a given time needs to be found.

E = mc squared

Mass and energy are two forms of the same thing — a tiny amount of mass converts into an enormous amount of energy, since the conversion factor (the speed of light squared) is such a huge number. This is exactly the energy source behind nuclear reactions.

E

— energy released, in joules (J)

m

— mass converted into energy, in kilograms (kg)

c

— the speed of light, 3 times 10 to the power 8 m/s

When to use it: Whenever the energy released by a nuclear reaction (fission or fusion) needs to be found from the mass converted during the reaction.

lambda = h / p = h / (mv)

Every moving particle, not just light, has an associated wavelength — even everyday objects, though their wavelength is far too tiny to ever notice.

lambda

— the particle's de Broglie wavelength, in metres (m)

h

— Planck's constant, 6.626 times 10 to the power -34 J times s

p

— the particle's momentum, in kilogram-metres per second (kg m/s)

m, v

— the particle's mass (kg) and speed (m/s)

When to use it: Whenever the wavelength associated with a moving particle (such as an electron) needs to be found.

delta t = gamma times delta t_0, where gamma = 1 / sqrt(1 minus v squared over c squared)

A clock moving at high speed relative to an observer runs slower, from that observer's point of view — time itself passes at a different rate depending on relative motion.

delta t

— the time interval measured by a stationary observer, in seconds (s)

delta t_0

— the "proper time" — the time interval measured by an observer moving WITH the clock, in seconds (s)

gamma

— the Lorentz factor (dimensionless), always greater than or equal to 1

v

— the relative speed between the two observers, in metres per second (m/s)

c

— the speed of light, 3 times 10 to the power 8 m/s

When to use it: Whenever comparing how much time passes for two observers in relative motion at a significant fraction of the speed of light.

n = m/M

The number of moles in a sample, from its mass and molar mass.

n

— amount of substance, in moles (mol)

m

— mass of the sample, in grams (g)

M

— molar mass, in grams per mole (g/mol) — numerically equal to Mr

When to use it: Whenever you need to convert between a measured mass and the number of moles it represents, or vice versa.

c = n/V

The concentration of a solution, from the moles of solute dissolved and the volume of solution.

c

— concentration, in moles per cubic decimetre (mol/dm³)

n

— amount of solute, in moles (mol)

V

— volume of solution, in cubic decimetres (dm³)

When to use it: Whenever you need to find how concentrated a solution is, or how many moles/what volume are needed to make one of a given concentration.

% yield = (actual yield / theoretical yield) × 100

How much of the maximum possible product a reaction actually produced, as a percentage.

actual yield

— the amount of product actually obtained

theoretical yield

— the maximum possible amount of product, predicted from the balanced equation

When to use it: Whenever you need to measure how efficient a reaction was at converting reactants into the desired product.

% atom economy = (desired product mass / total reactant mass) × 100

What percentage of the total reactant mass ends up in the desired product, rather than in by-products.

desired product mass

— the formula mass of the product you actually want

total reactant mass

— the combined formula mass of everything produced by the reaction, including by-products

When to use it: Whenever you need to measure how efficiently a reaction uses its atoms, especially when comparing different possible reaction routes.

ΔEN = |EN₁ − EN₂|

How unequally two bonded atoms share their electrons — the bigger the difference, the more one-sided the bond.

ΔEN

— electronegativity difference between the two bonded atoms (no units — the Pauling scale is a relative scale)

EN₁, EN₂

— the Pauling electronegativity values of each atom in the bond

When to use it: Whenever you need to predict what type of bond forms between two atoms: roughly ΔEN < 0.4 is nonpolar covalent, 0.4–1.7 is polar covalent, and ΔEN > 1.7 is ionic.

ΔH = H(products) − H(reactants)

The heat energy released or absorbed by a reaction at constant pressure.

ΔH

— enthalpy change of the reaction, in kilojoules per mole (kJ/mol)

H(products)

— total enthalpy (heat content) of the products

H(reactants)

— total enthalpy (heat content) of the reactants

When to use it: Whenever you need to classify a reaction as exothermic (ΔH negative) or endothermic (ΔH positive), or state how much heat energy a reaction releases or absorbs.

q = mcΔT

The heat energy absorbed or released by a substance, calculated from a measured temperature change — the standard way an enthalpy change is measured experimentally.

q

— heat energy transferred, in joules (J)

m

— mass of the substance (usually the water in a calorimeter), in grams (g)

c

— specific heat capacity of the substance, in J/(g·°C) — for water, c = 4.18 J/(g·°C)

ΔT

— change in temperature, in °C

When to use it: Whenever a reaction's heat is measured by tracking the temperature change it causes in a known mass of (usually water-based) surroundings.

ΔH(overall) = ΔH₁ + ΔH₂ + ΔH₃ + …

The overall enthalpy change for a reaction equals the sum of the enthalpy changes for any set of steps that add up to the same overall reaction.

ΔH(overall)

— enthalpy change for the reaction as a whole, in kJ/mol

ΔH₁, ΔH₂, …

— enthalpy changes of each individual step in an alternative pathway between the same starting reactants and final products

When to use it: Whenever a reaction's enthalpy change cannot be measured directly, but can be built from other reactions whose enthalpy changes are already known.

rate = Δ[concentration] / Δt

How quickly the concentration of a reactant or product changes over a time interval.

rate

— average reaction rate, typically in mol/(L·s)

Δ[concentration]

— change in concentration (mol/L) of a chosen reactant or product over the interval

Δt

— time interval over which the change is measured, in seconds (s)

When to use it: Whenever you have concentration measurements at two points in time and need the average rate of reaction between them.

k = A·e^(−Ea/RT)

How the rate constant of a reaction depends on temperature and activation energy — the mathematical basis for why raising temperature speeds up reactions so dramatically.

k

— the rate constant of the reaction

A

— the pre-exponential (frequency) factor — related to how often molecules collide with the correct orientation

Ea

— activation energy, in J/mol

R

— the gas constant, 8.314 J/(mol·K)

T

— absolute temperature, in kelvin (K)

When to use it: Whenever you need to explain or calculate how much a reaction rate changes with temperature, or compare the activation energies of different reactions.

For aA + bB ⇌ cC + dD: Kc = [C]ᶜ[D]ᵈ / [A]ᵃ[B]ᵇ

The ratio of product concentrations to reactant concentrations at equilibrium, each raised to the power of its coefficient in the balanced equation — a fixed value for a given reaction at a given temperature.

Kc

— the equilibrium constant (concentration-based), unitless in the way it is typically used here

[A], [B], [C], [D]

— equilibrium molar concentrations of each substance, in mol/L

a, b, c, d

— the balanced stoichiometric coefficients of each substance in the equation

When to use it: Whenever you need to quantify how far a reversible reaction proceeds toward products at equilibrium, or calculate an unknown equilibrium concentration.

Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ (at 25°C)

The product of hydrogen ion and hydroxide ion concentrations is always constant in water at a given temperature — true for pure water AND for any aqueous acid or base solution.

Kw

— the ionic product (self-ionisation constant) of water, 1.0 × 10⁻¹⁴ at 25°C

[H⁺]

— hydrogen ion (hydronium) concentration, in mol/L

[OH⁻]

— hydroxide ion concentration, in mol/L

When to use it: Whenever you know either [H⁺] or [OH⁻] in an aqueous solution at 25°C and need the other.

pH = −log₁₀[H⁺]

A convenient logarithmic scale for expressing how acidic or basic a solution is, compressing the enormous range of possible [H⁺] values into a simple 0–14 scale.

pH

— the pH value (no units)

[H⁺]

— hydrogen ion concentration, in mol/L

When to use it: Whenever you need to convert a hydrogen ion concentration into the standard pH scale, or vice versa (using [H⁺] = 10⁻ᵖᴴ).

pOH = −log₁₀[OH⁻] and pH + pOH = 14 (at 25°C)

pOH measures basicity the same way pH measures acidity — and because Kw links [H⁺] and [OH⁻], pH and pOH always add up to 14 at 25°C.

pOH

— the pOH value (no units)

[OH⁻]

— hydroxide ion concentration, in mol/L

pH

— the pH value of the same solution

When to use it: Whenever you know pH and need pOH (or [OH⁻]) without recalculating from scratch, at 25°C.

For HA ⇌ H⁺ + A⁻: Ka = [H⁺][A⁻] / [HA]

Quantifies how much a weak acid ionises in water — a larger Ka means a stronger (more ionised) weak acid.

Ka

— the acid dissociation constant, in mol/L

[H⁺]

— hydrogen ion concentration at equilibrium, in mol/L

[A⁻]

— concentration of the conjugate base at equilibrium, in mol/L

[HA]

— concentration of the undissociated weak acid at equilibrium, in mol/L

When to use it: Whenever you need to quantify or compare the strength of weak acids, or calculate the pH of a weak acid solution.

pH = pKa + log₁₀([A⁻] / [HA])

Calculates the pH of a buffer solution directly from the ratio of conjugate base to weak acid, without needing a full equilibrium calculation.

pH

— the pH of the buffer solution

pKa

— −log₁₀(Ka) of the weak acid used in the buffer

[A⁻]

— concentration of the conjugate base (e.g. from a salt) in the buffer, in mol/L

[HA]

— concentration of the weak acid in the buffer, in mol/L

When to use it: Whenever you need the pH of a buffer solution made from a weak acid and its conjugate base (or a weak base and its conjugate acid).

CₙH₂ₙ₊₂

Gives the molecular formula of any alkane (a saturated hydrocarbon with only single C–C bonds) directly from its number of carbon atoms.

n

— the number of carbon atoms in the molecule

CₙH₂ₙ₊₂

— the resulting molecular formula — n carbon atoms and (2n + 2) hydrogen atoms

When to use it: Whenever you know how many carbon atoms an alkane has and need its full molecular formula, or vice versa.

CₙH₂ₙ

Gives the molecular formula of any alkene (a hydrocarbon containing one C=C double bond) directly from its number of carbon atoms.

n

— the number of carbon atoms in the molecule

CₙH₂ₙ

— the resulting molecular formula — n carbon atoms and 2n hydrogen atoms

When to use it: Whenever you know how many carbon atoms an alkene (with one C=C double bond) has and need its full molecular formula.

Magnification = Image size / Actual size

How many times larger a microscope image is compared to the real, actual size of the object.

Image size

— the measured size of the image under the microscope

Actual size

— the object's real, true size

When to use it: Whenever you need to calculate how much a microscope image has been magnified, or work backwards to find an object's real size.