Problem:
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 be the region formed by the union of the square and all the triangles, and let be the smallest convex polygon that contains . What is the area of the region that is inside but outside ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The region inside but outside consists of four triangles, each of which has two sides of length 1. The angle between those two sides is . Thus the area of each triangle is
so the required area is .
The problems on this page are the property of the MAA's American Mathematics Competitions