Did Pythagoras actually prove the theorem named after him?

Probably not. The theorem was known in Babylon at least 1,000 years before Pythagoras, and clay tablets from around 1800 BCE list Pythagorean triples. The naming convention is a Greek tradition that attributes a result to whoever introduced it to the Greek mathematical community. Whether Pythagoras the person even existed is debated by historians.

Does it work for non-right triangles?

No. The Pythagorean theorem is a special case of the law of cosines, which works for any triangle. The general law is $c^2 = a^2 + b^2 - 2ab\cos C$, where $C$ is the angle opposite $c$. When $C = 90°$, $\cos C = 0$ and you recover the Pythagorean theorem.

What is a Pythagorean triple?

A set of three positive integers $a, b, c$ that satisfy $a^2 + b^2 = c^2$. The smallest is $(3, 4, 5)$. Others include $(5, 12, 13)$, $(8, 15, 17)$, $(7, 24, 25)$. There are infinitely many; Euclid gave a recipe for generating all of them.

Yes, the diagonal of a rectangular box with sides $a, b, c$ has length $\sqrt{a^2 + b^2 + c^2}$. The same pattern continues in any number of dimensions: the diagonal of an $n$-dimensional box is the square root of the sum of the squares of its sides.

Does it work on a sphere?

No, not in the same form. On a sphere, the relationship between the sides of a "right" triangle is $\cos c = \cos a \cdot \cos b$. The flat Pythagorean theorem is the limiting case as the triangle gets very small relative to the sphere's radius.

How many proofs of the Pythagorean theorem exist?

At least 400 documented. Loomis cataloged 370 in 1927; new ones (including Johnson and Jackson's 2023 trigonometric proof) keep appearing.

Visual proof · Flagship

The Pythagorean Theorem, Twelve Ways

One theorem. Twelve proofs. One page. From Bhāskara's 1150 dissection to a U.S. president's trapezoid to Einstein at twelve, every proof drawn, every insight named.

Quick answer

In any right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

Open this in math.im for the full interactive version →

Canonical URL: https://math.im/blog/pythagorean-theorem-12-proofs.html

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