The Pythagorean Theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². It is one of the most-used results in geometry, proven by Euclid and known for millennia before him to the Babylonians and Egyptians.

Practical Applications

Construction crews square corners with the 3-4-5 method; screen and TV sizes are quoted as the hypotenuse diagonal; GPS and surveying use it for distances; roofers and builders compute pitch and run with it daily.

Beyond Right Triangles

The Law of Cosines — c² = a² + b² − 2ab·cos C — extends the theorem to all triangles. When C = 90°, cos C = 0 and it collapses back to the Pythagorean theorem, which is why the two are taught as a pair.