2008 AIME II Problem 13

A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let R be the region outside the hexagon, and let S = \left\lbrace\frac{1}{z}|z \in R\right\rbrace. Then the area of S has the form a\pi + \sqrt{b}, where a and b are positive integers. Find a + b.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.