2004 AMC 12A Problem 24

A plane contains points A and B with AB = 1. Let S be the union of all disks of radius 1 in the plane that cover \overline{AB}. What is the area of S?

\text {(A) } 2\pi + \sqrt3 \qquad \text {(B) } \frac {8\pi}{3} \qquad \text {(C) } 3\pi - \frac {\sqrt3}{2} \qquad \text {(D) } \frac {10\pi}{3} - \sqrt3 \qquad \text {(E) }4\pi - 2\sqrt3

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