Question

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?

$\pi - \sqrt{3}$

$\pi - \sqrt{2}$

$\dfrac{\pi + \sqrt{2}}{2}$

$\dfrac{\pi + \sqrt{3}}{2}$

$\dfrac{7}{6}\pi - \dfrac{\sqrt{3}}{2}$

Solution:

The large semicircle has area $\dfrac12 \pi (2)^2 = 2\pi.$

Removing the small semicircles deletes a region equal to five congruent $60^\circ$ sectors of radius $1$ plus two equilateral triangles of side $1.$ Each sector has area $\dfrac{\pi}{6}$ and each triangle has area $\dfrac{\sqrt3}{4}.$

The shaded area is $2\pi - 5 \cdot \dfrac{\pi}{6} - 2 \cdot \dfrac{\sqrt3}{4} = \dfrac{7}{6}\pi - \dfrac{\sqrt3}{2}.$

Thus, the correct answer is E.

2003 AMC 10B Problem 19

Solution:

Problem 19 in Other Years