Calculus question with circle, and string tracing an area The figure shows a piece of string tied to a circle with a radius of one unit. The string is just long enough to reach the opposite side of the circle. Find the area of the region, not including the circle itself that is traced out when the string is unwound counterclockwise and continues counterclockwise until it reaches the opposite side again.
I'm mostly condensing what Eric Weisstein wrote in this MathWorld article, citing a 1998 paper by Hoffman, The Bull and the Silo: An Application of Curvature. You can solve this by finding the area of the semicircle the string traces (before the string begins to wrap) and then do some fancy calculus to find the area of the regions formed when the string begins to wrap around the circle. 

By the calculations in the article, given a circle of radius $a$ and a string of length $L$, the area $A$ will be
$$A = \frac{\pi L^2}{2} + \frac{L^3}{3a}\;.$$

So in your case where $a=1$ and $L=\pi$ we have $A = \frac{5}{6}\pi^3 \;\approx\; 25.838564$.
Well, I cheated, and solved it numerically: it comes up at about 19.55.

Unfortunately that does not match up with Mike Pierce's answer of $\frac56 \pi$...

Here is a graph of the interesting part, and here is the Matlab code.