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2003 AMC 12B Question 16

AMC12 · 2003 AMC 12B

Question 16

Three semicircles of radius $1$ are constructed on diameter $\overline{AB}$ of a semicircle of radius $2.$ The centers of the small semicircles divide $\overline{AB}$ 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?

Question illustration
  1. $\pi - \sqrt{3}$
  2. $\pi - \sqrt{2}$
  3. $\dfrac{\pi + \sqrt{2}}{2}$
  4. $\dfrac{\pi + \sqrt{3}}{2}$
  5. $\dfrac{7}{6}\pi - \dfrac{\sqrt{3}}{2}$