The Mysterious Square

When we rearrange the pieces of the upper picture to form the lower picture we have a missing square. The pieces in both pictures are identical. Can you explain why the square is missing? You will need your knowledge of geometry to solve this problem.

Answer

The area of a right triangle is computed by multiplying the base times the height and dividing by two. The pieces of the puzzle have an area of 32 square units. Although they can be assembled to form what appears to be a 13x5 right triangle, they actually form quadrilaterals that are slightly smaller or slightly bigger than a 13x5 right triangle. The long side of the these triangles, i.e, the hypotenuse, is not a straight line. The top figure has an area of 32 square units. The bottom figure, including the empty square, has an area of 33 square units. A real 13x5 right triangle would have an area of 32.5 square units. The distortion is difficult to see because one square of the picture is approximately 3% of the area. The distortion can be seen more clearly when the empty square constitutes a larger percentage of the area, as in the figure below where 1 square represents 13% of the area.