Today, I was browsing through StumbleUpon, and I ran across this popular illusion that has been around for a while. The illusion is, that there are four shapes (as shown),

that when arranged differently form the same shape, but with a different Area.

Comparing the upper and lower triangles (we’ll call them X and Y, respectively), we can see that X has an empty square that seemingly disappears when rearranged into Figure Y, yet at a glance, the size of X and Y in their entirety seem to be identical. However, since the area of the individual shapes have not changed, this simply cannot be possible! So let’s break it down and see exactly what is happening in between arrangements.

Our goal here is to prove that the area of triangles X and Y are *not* equal. There are a few different ways that we can do this, but what we will do in this case is calculate areas A and B outside of the triangle. If these triangles are, in fact, identical, then areas A and B should be exactly equal.

We’ll calculate this with the standard formulas for area:

- Rectangle: A = bh
- Triangle: A = 1/2bh

We’re interested in the total area of the colored shapes above. We now have our totals for both Areas A and B. We can see that in triangle X, where there is the unexplained square, the outer area “A” is 32. In triangle Y, outer area “B” is 33, the difference of 1 square inch caused by the extra space in X.

Why this illusion is so effective at first glance, is that our eyes assume that the hypotenuse of X or Y as a whole is a straight line. In actuality, the slopes of the two aligning triangles are slightly different. In X, the line bends out slightly outwards, while in Y, the line retracts inward. Just enough to make that 1 square inch difference.

This dose of geekery brought to you by 2:30 AM, and the addictive power of StumbleUpon. If you found this informative, please Stumble it. :)