# 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
$90\xb0$
angle.

The
converse
of the theorem is also true: if a triangle has sides of lengths
$a,b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}c$
, and
${c}^{2}={a}^{2}+{b}^{2}$
, then it must be a
**
right
**
triangle
*
.
*

See also Pythagorean Triples .