What Is the Pythagorean Theorem

Last updated: March 31, 2026

Quick Answer: In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse (c) is the longest side, opposite the right angle.

Key Facts

Named after Pythagoras (~570-495 BC) but Babylonians knew it 1,000 years earlier
Simplest example: 3² + 4² = 25 = 5²
Over 400 known proofs exist
Only works for right-angled triangles
Extends to 3D as d = sqrt(x² + y² + z²)

The Formula

a² + b² = c² where c is always the hypotenuse (longest side, opposite the 90° angle).

Example

Legs of 3 and 4: 9 + 16 = 25 → hypotenuse = 5. Common triples: 3-4-5, 5-12-13, 8-15-17.

Real-World Uses

Construction: 3-4-5 rule verifies right angles
Navigation: Shortest distance between points
Computer graphics: Distance calculations in 2D/3D
GPS: Trilateration uses extended form

History

Babylonians used it c. 1800 BC. Ancient Indian, Chinese, and Egyptian mathematicians discovered it independently. Pythagoras credited with first known proof.

Sources

Wikipedia — Pythagorean Theorem CC-BY-SA-4.0