Special Right Triangles (45-45-90 and 30-60-90)
Simple Explanation
Two right triangles have fixed, memorizable side ratios: a 45-45-90 triangle (an isosceles right triangle) has sides in the ratio x : x : x√2, and a 30-60-90 triangle has sides in the ratio x : x√3 : 2x.
Why Do We Need It?
Knowing these ratios lets you find every side of these specific, very common triangles from just one known length, instantly, without needing the Pythagorean theorem each time.
See It
An isosceles right triangle with two equal legs of length x and a hypotenuse of length x times the square root of 2
Formula
Special Right Triangle Side Ratios
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.
Worked Example
Use the 30-60-90 ratio
A 30-60-90 triangle has its shortest side (opposite the 30° angle) equal to 5. Find the other two sides.
Why Does This Work?
For 45-45-90: the two legs are equal (isosceles), so if each leg is x, the Pythagorean theorem gives hypotenuse = √(x²+x²) = x√2. For 30-60-90: this triangle is exactly half of an equilateral triangle (cut along an altitude) — the equilateral side becomes the hypotenuse (2x), half its base becomes the short leg (x), and the altitude (found via the Pythagorean theorem) becomes the long leg, x√3.
Real-Life Example
Roof pitch and ramp angles
A roof truss or wheelchair ramp built at a standard 30° or 45° angle forms one of these special right triangles.
Builders use the fixed side ratios to instantly calculate material lengths for a given height or base, without recomputing the Pythagorean theorem for every job.
Practice
A 45-45-90 triangle has a leg of length 7. Find the hypotenuse. (Give as a decimal, 2 d.p.)
MediumCommon mistake
Mixing up which side gets multiplied by √3 in a 30-60-90 triangle — it is always the LONG leg (opposite the 60° angle) that equals x√3; the hypotenuse is simply 2x, with no square root involved.
Quick Review
- 45-45-90: legs x, x; hypotenuse x√2.
- 30-60-90: short leg x; long leg x√3; hypotenuse 2x.
- A 30-60-90 triangle is half of an equilateral triangle, split along its altitude.