The Triangle Dissection Fallacy

Take a look at the above picture, and agree with me that this is weird. We see two triangles, and based on the grid they are both the same size. A quick count of the grid boxes tells us that the surface of both triangles should be 13 × 5 ÷ 2. All the pieces are equal in size, yet by shuffling them around we suddenly have a spare grid box!

This does not make any sense. The act of rearranging the triangle pieces should not change the surface area. What's going on here?

Solution

Take a closer look at the picture. You are being fooled, but the effect is so small, that we might as well call this an optical illusion.

In the above picture I have overlayed the big triangles on top of each other. As you can see, they are not exactly identical after all! The reason is that the slope of the red and blue piece do not match. Let's confirm.
Red piece angle: tan^-1(3 ÷ 8) = 20.6°
Blue piece angle: tan^-1(2 ÷ 5) = 21.8°
They're close, but not the same, so the big triangles can never be right triangles. They're actually polygons with four sides.

That little area in the overlay that is different between the triangles polygons? It's all stretched out there, but squeeze it into a square and you'll see that it exactly matches one grid box. So that's were that went!