visual proof of the pythagorean theorem

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The pythagorean theorem, also known as the hecatomb theorem or even at the Byzantium era, the skadra rule (skadra means square in Greek).

In any right-angled triangle, the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the squares whose sides are the two legs (the two sides that meet at a right angle)

Let's begin with a right angle square (side α is the hypotenuse)

What does the theorem states? The square  of the hypotenuse... so we will create a square whose side is equal to the hypotenuse. We make a square with side length β+γ.

We got four same triangles, we rotated them in a proper way and so in the inner  a new square appears. This square has side length equal to the hypotenuse. Nice, at this point let's make our shape a bit more colorful.

At this step, nothing has changed.Isn't that right? Well, now we will fold the triangles inside the second square.

We filled the second square with four same triangles and as it's obvious there is a third square leftover in the middle. We should do something with that, or maybe not... Still the sum of the squares of the two legs hasn't appeared. Observe the shape, we are pretty close.

Can you see them?
In green color there is a square with side length β
and with blue a square with side length  γ.
Let's see what happens if we we move the red and the yellow triangle. Be patient, we almost done!

The outline that the squares of the legs form, consists by four same triangles and in the middle the same square that was leftover before.Check again above, isn't that what we did to fill up the hypotenuse square? We made it! We have confirmed the pythagorean theorem, in a simple, non mathematical hard, way.

Tip: In order, the steps we followed, to be better understood you can use paper and cut the respective pieces. It will help a lot, specially the younger kids.