Below is a visual proof (!) that $32.5 = 31.5$. How could that be?



(As noted in a comment and answer, this is known as the "Missing Square" puzzle.)

It's an optical illusion - neither the first nor second set of blocks actually describes a triangle. The diagonal edge of the first is slightly concave and that of the second is slightly convex.

To see clearly, look at the gradients of the hypotenuses of the red and blue triangles - they're not 'similar'.

gradient of blue triangle hypotenuse = 2/5
gradient of red triangle hypotenuse = 3/8  

Since these gradients are different, combining them in the ways shown in the diagram does not produce an overall straight (diagonal) line.
