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 $S$? Express your answer in terms of $\pi$ and in simplest radical form.
Question:
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 $S$? Express your answer in terms of $\pi$ and in simplest radical form.
Answer:
Let's think step by step.

user
3\pi-\frac{\sqrt{3}}{2}
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 $S$? Express your answer in terms of $\pi$ and in simplest radical form.
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