how does the "Missing Square illusion" work?

Question

When the shapes are rearanged where does the square go?
http://www.coolopticalillusions.com/illusions/missingsquare.gif

Answer 1

The snag is that these four shapes seem to form a triangle together, but it isn't a real triangle! Compute the areas of the shapes:

Yellow: 7
Green: 8
Red: 12
Blue: 5
Total: 32

Together they seem to form a triangle with an area of (5 * 13) / 2 = 32.5

In the first case, the shapes together form an area of 32, which is 0.5 less than the area of triangle they seem to form. In the second case, they form an area of 33, which is 0.5 more. So the difference in area is 1, and that's the "hole" in the second area

The tricky part is the hypotenuse of the "virtual triangle". The steepness of the hypotenuse of the red triangle is 3 / 8 (= 15/40), that of the blue one is 2 / 5 (= 16/40). So the one of the blue triangle is a very little bit steeper. You almost don't see it, especially if you draw them on a grid...or can you see it now?

The two "triangles" are in fact not triangles at all, but quadrilaterals, because the "hypotenuse" is not quite straight.

In the first case you have a concave quadrilateral that bends *in* slightly. In the second case you have a convex quadrilateral that bends *out* slightly.

If the two quadrilaterals are drawn on top of each other, you can see the area of difference (which is actually a narrow parallelogram). The area of this additional parallelogram is found to be exactly one square unit. That accounts for the "missing" square.

See the picture below...

Answer 2

It's an optical illusion. It seems that both triangles (red and blue) are similar (equal internal angles) but they're not.
The top figure has an area of 32 square units. The bottom figure (including the empty square) has an area of 33 square units.