Pythagorean Theorem Explorer

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

You can solve for any side. Given two sides $a$ and $b$, the hypotenuse is $c = \sqrt{a^2 + b^2}$. Given one leg and the hypotenuse, the other leg is $a = \sqrt{c^2 - b^2}$.

Draw a square on each side of a right triangle. The area of the square on the hypotenuse equals the combined area of the squares on the two legs.

For the classic 3-4-5 triangle, $9 + 16 = 25$. This geometric interpretation is why the theorem works: it is fundamentally a statement about areas.

A Pythagorean triple is a set of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. A triple is primitive if $\gcd(a, b, c) = 1$.

Euclid's formula generates all primitive triples when $m > n > 0$, $\gcd(m, n) = 1$, and $m - n$ is odd. Multiply by any positive integer $k$ to get derived triples.

The Pythagorean theorem extends to three dimensions. The distance from the origin to $(x, y, z)$ uses the theorem twice.

First find the distance in the xy-plane: $d_{xy} = \sqrt{x^2 + y^2}$. Then combine with z: $d = \sqrt{d_{xy}^2 + z^2}$. Each step is one application of $a^2 + b^2 = c^2$.