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?
Answer Choices:
A.
B.
C.
D.
E.
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
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The problems on this page are the property of the MAA's American Mathematics Competitions