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The Pythagorean Theorem

The Pythagorean Theorem is named after Pythagoras of Samos , a mathematician who was also a religious leader, and believed that all things in the universe were composed of numbers.

He is supposed to have been the first to have proved this theorem about right triangles:

Pythagorean Theorem. In a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ , the following equation is true:

$c^2=a^2+b^2$

(There are many different ways to prove this.)

The hypotenuse of a right triangle is the side opposite the right angle.

Below is a graphical representation. The theorem states that the sum of the areas of the blue and red squares is equal to the area of the green square.

Important : Remember that the Pythagorean Theorem is true only for right triangles – triangles which have a $90°$ angle.

The converse of the theorem is also true: if a triangle has sides of lengths $a,b,\text{and}c$ , and $c^2=a^2+b^2$ , then it must be a right triangle .

See also Pythagorean Triples .