The Pythagorean Theorem

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

This single line of math underpins surveying, navigation, construction, and the distance formula in coordinate geometry.

Using the theorem

Example: legs of 3 and 4. What is the hypotenuse?

1. Square the legs: 3² = 9 and 4² = 16
2. Add them: 9 + 16 = 25
3. Take the square root: √25 = 5

So a 3-4-5 triangle is right-angled. Builders use this exact triple (and its multiples like 6-8-10) to check that corners are square.

Working backwards

The theorem also finds a leg when you know the hypotenuse:

b² = c² − a²

Example: hypotenuse 13, one leg 5. Then b² = 169 − 25 = 144, so b = 12. The 5-12-13 triangle is another classic Pythagorean triple.

Common triples worth memorizing

• 3, 4, 5 (and 6-8-10, 9-12-15, 30-40-50…)
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25

A note on irrational results

Most triangles do not produce a whole number. Legs of 1 and 1 give a hypotenuse of √2 ≈ 1.414… — an irrational number. This discovery is said to have shocked the ancient Pythagoreans, who believed all lengths were rational.

When not to use it

The theorem applies only to right triangles. For oblique triangles you need the Law of Cosines, which is its generalization: c² = a² + b² − 2ab·cos(C). When C = 90°, cos(C) = 0 and you are back to the Pythagorean theorem.