Pythagorean Theorem Proofs — Euclid's and Garfield's, Step by Step

The Pythagorean Theorem: Proofs, the Way They Were First Written

In a right triangle, the square on the longest side (the hypotenuse) has the same area as the squares on the two other sides together: a² + b² = c². The rule was known in Babylon, India and China long before Euclid; Euclid's Elements gives the classical proof, and hundreds of other proofs exist — one by a future US President.

In a right triangle with legs a and b and hypotenuse c:

a ² + b ² = c ²

The rule is older than Pythagoras — Babylonian tablets, Indian altar manuals and Chinese texts know it — and there are hundreds of proofs. On this site we follow each proof from its original source, step by step, with the figure as it was first drawn.

Two proofs to start with:

Euclid (around 300 BC) proves it with areas: the big square splits into two rectangles, each equal to one of the small squares. No numbers, no algebra.
James A. Garfield (1876), then a congressman and later the 20th US President, counts the area of one trapezoid in two ways.

The Pythagorean Theorem: Proofs, the Way They Were First Written

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