2003 AMC 12B Problems/Problem 16

Problem

Three semicircles of radius $1$ are constructed on diameter $AB$ of a semicircle of radius $2$ . The centers of the small semicircles divide $AB$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?

$\textbf{(A) } \pi - \sqrt{3} \qquad\textbf{(B) } \pi - \sqrt{2} \qquad\textbf{(C) } \frac{\pi + \sqrt{2}}{2} \qquad\textbf{(D) } \frac{\pi +\sqrt{3}}{2} \qquad\textbf{(E) } \frac{7}{6}\pi - \frac{\sqrt{3}}{2}$

Solution

Each small semicircle shares a radius with an adjacent circle. Therefore, the radii to the points of intersection will create equilateral triangles. Draw these triangles, and then bisect the $120^\circ$ angles on the sides and complete the incomplete triangles so that the unshaded region is broken into two types of pieces: circular segments from a $60^\circ$ sector of a circle with radius $1$ , and equilateral triangles with side length $1$ .

There are $7$ equilateral triangles and $5$ circular segments. One equilateral triangle has area $\frac{1}{2}(1)\left( \frac{\sqrt{3}}{2} \right)=\frac{\sqrt{3}}{4}$ , and one segment as area $\frac{60}{360}\pi (1^2)-\frac{\sqrt{3}}{4}=\frac{\pi}{6}-\frac{\sqrt{3}}{4}$ .

So the shaded region is the area of the large semicircle minus the area of the unshaded parts, which is $\frac{1}{2}\pi (2^2) - 7\left( \frac{\sqrt{3}}{4}\right) - 5\left( \frac{\pi}{6}-\frac{\sqrt{3}}{4}\right) = 2\pi - \frac{7\sqrt{3}}{4} - \frac{5\pi}{6} - \frac{5\sqrt{3}}{4}= \boxed{\textbf{(E) } \frac{7}{6}\pi - \frac{\sqrt{3}}{2}}$ .

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2003 AMC 12B Problems/Problem 16

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