False Proof—31.5 = 32.5

Problem: Show 31.5 = 32.5. “Solution”: Explanation: It appears that by shifting around the pieces of one triangle, we have constructed a second figure which covers less area! Since the first triangle has base length 13 and height 5, its area is 32.5. Clearly, the second figure has the same area minus a square of area 1, giving the second figure area 31.5. In fact, by counting up the squares in each component, we do see that it is simply a shifting of pieces, so the areas of the two figures must ...