Let $ABC$ be an equilateral triangle with side length $2$. Let $S$ be the union of all disks of radius $1$ whose centers lie inside or on $\triangle ABC$. What is the area of $S$?

1. The region consists of $\triangle ABC$, three rectangular strips of dimensions $2\times1$, and three $120^\circ$ sectors of radius $1$.
2. The triangle has area $\sqrt3$, the strips total area $6$, and the sectors total area $\pi$.
3. Therefore, $[S]=\boxed{6+\sqrt3+\pi}$.