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

$\dfrac{1}{4}$

$\dfrac{\sqrt{2}}{4}$

$1$

$\sqrt{3}$

$2\sqrt{3}$

Solution:

The convex hull $S$ differs from $R$ only near the four corners of the square, where a small triangular gap forms. Each gap triangle has two sides of length $1$ (outer edges of adjacent triangles).

The angle between those two sides is $360^\circ - 90^\circ - 4 \cdot 60^\circ = 30^\circ,$ so each gap has area $\tfrac12 \cdot 1 \cdot 1 \cdot \sin 30^\circ = \tfrac14.$

The total area is $4 \cdot \tfrac14 = 1.$

Thus, the correct answer is C.

2008 AMC 12B Problem 15

Solution:

Problem 15 in Other Years