Pythagoras' Theorem: The Most Famous Equation in Geometry

Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². It's elegant, simple, and useful in a way that very few theorems manage to be. Given any two sides of a right-angled triangle, you can always find the third.

Older than Pythagoras

Despite the name, the theorem was known well before the Greek mathematician Pythagoras of Samos (around 570–495 BC). A Babylonian clay tablet called Plimpton 322, dating to around 1800 BC, lists what appear to be Pythagorean triples — integer sets satisfying a² + b² = c² — suggesting the relationship was known and used at least 1,200 years before Pythagoras. Ancient Indian mathematical texts (the Sulbasutras, around 800–600 BC) describe the same relationship for constructing fire altars. What Pythagoras — or his school — may have contributed was the first formal proof that the theorem holds for all right-angled triangles, not just specific cases.

Used in almost every field involving distance

Builders use Pythagoras' theorem to check that corners are square — a classic 3-4-5 triangle confirms a right angle without a protractor. Surveyors use it to calculate distances across terrain. GPS receivers use it (in 3D form, extended to three dimensions) to calculate your position from satellite distances. A TV advertised as '65 inches' is 65 inches along its diagonal — computed using the theorem from the width and height. Sports field markers use it to lay out perfectly rectangular pitches. Any time you know two sides of a right triangle and need the third, Pythagoras is there.