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 $\dfrac{1}{2}\pi(2)^2 = 2\pi.$

Where the small semicircles overlap, adjacent ones meet at points a distance $1$ from two centers, forming equilateral triangles. The region removed from the large semicircle consists of five congruent $60^\circ$ sectors of radius $1,$ each of area $\dfrac{\pi}{6},$ together with two equilateral triangles of side $1,$ each of area $\dfrac{\sqrt{3}}{4}.$

The shaded area is $2\pi - 5\cdot\frac{\pi}{6} - 2\cdot\frac{\sqrt{3}}{4} = \frac{7}{6}\pi - \frac{\sqrt{3}}{2}.$

Thus, the correct answer is E.

2003 AMC 12B Problem 16

Solution:

Problem 16 in Other Years