AMC 12 2004a 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 $S$?

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