# Self-Diagram

A few weeks ago, I stumbled across this gem of a geometry problem from an old MATHCOUNTS exam:

The intersection of a circular region of radius $3$ inches and a circular region of radius $2$ inches has area $\pi$ square inches. In square inches, what is the area of the total region covered by the two circular regions?

(MATHCOUNTS Chapter Countdown 2013, Problem 8)

This problem is straightforward: the familiar $\pi r^2$ formula tells us that the two circles have area $9\pi$ and $4\pi$, and since the intersection takes up $\pi$ of those areas, the remaining regions have area $8\pi$ and $3\pi$. The total area is $8\pi+\pi+3\pi=\boxed{12\pi}$.

What makes this simple little problem so delightful? With the proper labels, it becomes its own Venn diagram, which amuses me more than it should.

If you are also tickled by this sort of thing, check out this relevant xkcd strip.