smc

Question

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 S be the smallest convex polygon that contains R. What is the area of the region that is inside S but outside R?

Accepted Answer

FIGURE The equilateral triangles form trapezoids with side lengths 1, 1, 1, 2 (half a unit hexagon) on each face of the unit square. The four triangles "in between" these trapezoids are isosceles triangles with two side lengths 1 and an angle of 30^ in between them, so the total area of these triangles (which is the area of S - R) is, 4 ( 12 30^ ) = 1 which makes the answer 1.