A regular hexagon of side length $1$ has a square of side length $1$ constructed outward on each of its six sides. Let $R$ be the union of the hexagon and the six squares, and let $S$ be the smallest convex polygon containing $R$. What is the area inside $S$ but outside $R$?

1. The convex hull adds one triangular gap at each hexagon vertex. Each gap has two sides of length $1$ and included angle $60^\circ$.
2. Each gap has area $\frac12(1)(1)\sin60^\circ=\frac{\sqrt3}{4}$, so the six gaps have total area $\boxed{\frac{3\sqrt3}{2}}$.