What Is the Pythagorean Theorem? a² + b² = c² Explained

Q: What is the Pythagorean theorem?

The Pythagorean theorem says that in a right triangle, the square of the longest side, the hypotenuse, equals the sum of the squares of the other two sides: a² + b² = c². It’s named after the Greek mathematician Pythagoras (about 570–495 BC), although Babylonian and other ancient mathematicians knew the relationship much earlier.

A worked example

A right triangle has sides of 3 cm and 4 cm. How long is the hypotenuse?

a² + b² = c²
3² + 4² = c²
9 + 16 = 25
c = √25 = 5 cm

Pythagorean triples

Whole numbers that fit the formula are called Pythagorean triples:

a	b	c
3	4	5
5	12	13
8	15	17
7	24	25

Builders have long used a rope with knots in a 3-4-5 pattern to make perfect right angles.

Who was Pythagoras?

Pythagoras was born on the Greek island of Samos and later founded a community of followers in Croton, southern Italy. The Pythagoreans believed that numbers explained the universe, from music to the stars. They also followed strict rules, reportedly including avoiding beans. Because he left no writings, historians aren’t sure which discoveries were his own.

It’s older than Pythagoras

A Babylonian clay tablet called Plimpton 322, from around 1800 BC, lists numbers that form Pythagorean triples.
Ancient Indian texts, the Sulba Sutras, and Chinese mathematicians also described the rule.
The first known written proof in the Greek tradition appears in Euclid’s Elements, around 300 BC. Euclid worked in Alexandria, home of the famous ancient library.

There are now hundreds of known proofs, including one by US President James Garfield in 1876.

Where it’s used

Field	Example
Construction	Checking corners are square
Navigation and GPS	Calculating straight-line distances; see what GPS stands for
Screens	A TV’s size is its diagonal
Video games	Measuring distance between objects

For a shape with one hole, see what a doughnut shape is called in math.