The Pythagorean theorem can be proven in many different ways. In this article we will show you one of these proofs of Pythagoras.

The theorem states that in a right triangle the square on the hypotenuse equals to the sum of the squares on the two legs.

### Proof of the Pythagorean theorem

This proof is a bit like a puzzle where we use a square with a lot of right triangles.

The area $A$ of the blue big square is

$ A = (a+b)·(a+b) $

We can also express the area $A$ by using the small pieces (the smaller square and the four right triangles)

$A= c^2+4·\frac{a·b}{2} $

(The smaller square and the four right triangles that all has the same area)

Let´s simplify this by dividing $ \frac42 = 2 $

$A= c^2+4·\frac{a·b}{2} = c^2 + 2ab $

The both areas has to be equal so we get

$(a+b)·(a+b) = c^2 + 2ab$

Expanding $ (a+b)·(a+b)=a^2+2ab+b^2 $

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

Subtract both sides with $2ab$

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

And we are done with the proof! By using this geometric “puzzle” we have proven that the square $c$ equals the sum of the square of the two legs $a$ and $b$.