Pythagorean theorem

A right triangle with the squares of each side shown.

The Pythagorean theorem posits that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of both legs. This is perhaps one of the most oft-proven theorems. The Pythagorean Proposition, a book published in 1940, contains 370 proofs of Pythagoras' theorem, including one by American President James Garfield. Making the Pythagorean theorem useful for finding distance in 2-coordinate graphs and diagonals of rectangles.

Pythagorean theorem

Let $\triangle ABC$ be a triangle.

If $\triangle ABC$ is a right triangle with the right angle at $B$ , then

$AB^{2}+BC^{2}=AC^{2}$

Conversely, if $AB^{2}+BC^{2}=AC^{2}$ , then $\triangle ABC$ is a right triangle with the right angle at $B$ .

This is more commonly stated mathematically:

$a^{2}+b^{2}=c^{2}$ , assuming that $c$ is the length of the side opposite the right angle (the hypotenuse).

See also

• Pythagorean theorem/Proof
• Pythagorean Theorem in the n-dimensional space
• Parallelogram law