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jmc

geometry senior

Problem

Three semicircles of radius 1 are constructed on diameter of a semicircle of radius 2. 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? Express your answer in terms of and in simplest radical form.

problem
Solution
The area of the larger semicircle is The region deleted from the larger semicircle consists of five congruent sectors and two equilateral triangles. The area of each of the sectors is and the area of each triangle is so the area of the shaded region is
Final answer
\frac{7}{6}\pi - \frac{\sqrt{3}}{2}