The Pythagoras Theorem
Simple Explanation
In any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c².
Why Do We Need It?
This is one of the most-used results in all of mathematics — it connects the three sides of any right triangle, letting you find any one side from the other two.
See It
A right triangle with legs of length 3 and 4 and hypotenuse of length 5, with the right angle marked
Formula
The Pythagorean Theorem
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.
Worked Example
Find the hypotenuse of a right triangle
A right triangle has legs of length 6 and 8. Find the hypotenuse.
Why Does This Work?
Stated simply, using similar triangles (connecting back to this chapter's theme): drop an altitude from the right angle to the hypotenuse, splitting the triangle into two smaller triangles, each similar to the original (by AA — they share an acute angle and both have a right angle). Writing the resulting similarity ratios and adding them together algebraically produces exactly a² + b² = c².
Real-Life Example
Checking a square corner in construction
Builders check that a wall meets the floor at a true right angle by measuring 3 units along one edge, 4 units along the other, and confirming the diagonal is exactly 5 units.
This "3-4-5 rule" is a direct, practical application of the Pythagorean theorem to verify a right angle without a protractor.
Practice
A right triangle has a leg of 5 and a hypotenuse of 13. Find the other leg.
MediumCommon mistake
Applying a² + b² = c² to a triangle that is not a right triangle — this theorem only holds for right triangles, and c must specifically be the side opposite the right angle.
Quick Review
- a² + b² = c², where c is the hypotenuse (opposite the right angle).
- Proof idea: an altitude to the hypotenuse creates two smaller similar triangles whose ratios combine into this result.
- Only applies to right triangles.