This is a simple manipulative that shows how the Pythagorean Theorem can be proven using rearrangement.

We start with a tray with a fixed internal area equal to (a+b) squared and four identical triangles with sides of length a, b and c.

If we arrange the triangles around the perimeter of the tray, as shown in figure one, the negative space (space not occupied by the triangles) is a square with sides of length c.

When we rearrange the triangles to form the rectangles, as shown in figure two, the negative space is split between two squares, one with sides of length "a" and a second with sides of length “b.”

Because the negative space cannot change, this proves that a2 + b2 = c2.

This thing was made with Tinkercad. Edit it online https://www.tinkercad.com/things/8ba9Bfx64fZ

Originally posted on Thingiverse.