2003 AMC 12B Problems/Problem 16
Problem
Three semicircles of radius are constructed on diameter of a semicircle of radius . The centers of the small semicircles divide 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?
Solution
Each small semicircle shares a radius with an adjacent circle. Therefore, the radii drawn to the points of intersection will create equilateral triangles. Bisect the angles on the sides and complete the incomplete triangles with lines so that the unshaded region is broken into two types of pieces: a circular segment from a sector of radius , and an equilateral triangle with side length .
There are equilateral triangles and circular segments. One equilateral triangle has area , and one segment as area .
So the shaded region is the area of the large semicircle minus the area of the unshaded parts, which is .