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2003 AMC 10B Problem 19

Problem 19 of 25HarderGeometry

Three semicircles of radius 11 are constructed on diameter AB‾\overline{AB} of a semicircle of radius 2.2. The centers of the small semicircles divide AB‾\overline{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?

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Solution

The large semicircle has area 12π(2)2=2π.\dfrac12 \pi (2)^2 = 2\pi. Removing the small semicircles deletes a region equal to five congruent 60∘60^\circ sectors of radius 11 plus two equilateral triangles of side 1.1. Each sector has area π6\dfrac{\pi}{6} and each triangle has area 34.\dfrac{\sqrt3}{4}. The shaded area is 2π−5⋅π6−2⋅34=76π−32. \begin{gathered} 2\pi - 5 \cdot \dfrac{\pi}{6} - 2 \cdot \dfrac{\sqrt3}{4} \\ = \dfrac{7}{6}\pi - \dfrac{\sqrt3}{2}. \end{gathered} Thus, the correct answer is E.

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Concepts: circle area · sector · equilateral triangle · area decomposition

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.