I ran across a very interesting question a few days ago. Given a pentagon that has side lengths 5, 5, 5, 6, and 8, what is the largest area the pentagon can enclose?

The answer is 3*(25*sqrt(3)/4) + 12 + 12 = 24 + 75*sqrt(3)/4, or approximately 56.476.

I'll post the reasoning if anyone is interested. Even just finishing off the last bit (as I did just now) was interesting. Whoever came up with this question chose it very carefully; what a wonderful problem. As a hint, I used the fact that for a given perimeter, the largest area you can enclose is enclosed by a circle. [Formally, this is known as the isoperimetric inequality.]