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:

c² = a² + b²

(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^o angle.

The converse of the theorem is also true: if a triangle has sides of lengths a, b, and c, and c² = a² + b², then it must be a right triangle.

See also Pythagorean Triples.