If I were to show you this tiling pattern I doubt that you would think it had anything to do with Pythagoras’ theorem.

However on further investigation we can see how it was constructed.

This is a small section of the grid and when we use different colours to fill it in it starts to make more sense.

In the figure below, the blue square is the square of the hypotenuse of the red triangle.  The blue square is clearly equal in area to the purple plus the green square.  But the purple square is the square of one of the legs of the red triangle.  The green square is the square of the other leg.  This dissection is attributed to Henry Perigal.

You can create your own Pythagorean tiling pattern. Firstly;

Start with any right triangle

Add the square on the hypoteneuse

Add the square on one sideExtend the sides to start

the gridNow the pattern has been established repeat it:

The red and yellow squares are the squares on the legs.  These 2 squares can be cut up and reassembled into a square that is the same size as the square on the hypotenuse.  This dissection was devised by Thabit ibn Qurra around 900 A.D.