Pythagorean Theorem

The Pythagorean Theorem (Pythagoras' theorem) is a fundamental relation between the three sides of a right triangle. It states that the square of the hypotenuse (the longest side, opposite the 90° angle) equals the sum of the squares of the other two sides. Given a right triangle with sides a, b, and hypotenuse c: a² + b² = c².

This relationship allows determining the third side when two are known. The law of cosines generalizes the Pythagorean theorem to any triangle; when the included angle is 90°, it reduces to the Pythagorean equation. There are numerous proofs of the theorem, including algebraic proofs by rearranging areas.