Question 15
On each side of a unit square, an equilateral triangle of side length $1$ is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length $1$ is constructed. The interiors of the square and the $12$ triangles have no points in common. Let $R$ be the region formed by the union of the square and all the triangles, and let $S$ be the smallest convex polygon that contains $R.$ What is the area of the region that is inside $S$ but outside $R?$