Proof Pythagorean theorem


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.

the pythagorean theorem

Proof of the Pythagorean theorem

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

Proof pythagorean theorem

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 $b$.