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?

$\pi - \sqrt{3}$

$\pi - \sqrt{2}$

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

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

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

Answer: E

Concepts:sector equilateral triangle area decomposition

Difficulty rating: 1680

Solution:

The large semicircle has area $\dfrac{1}{2}\pi(2)^2 = 2\pi.$

The three small semicircles have total area $\dfrac{3\pi}{2}$ before their overlaps are accounted for. The centers of each adjacent pair are $1$ unit apart, so their intersection is bounded by two $60^\circ$ sectors and an equilateral triangle. Each of the two overlaps therefore has area $\dfrac{\pi}{3} - \dfrac{\sqrt{3}}{4}.$

The shaded area is $\begin{aligned} &2\pi - \left[\frac{3\pi}{2} - 2\left(\frac{\pi}{3} - \frac{\sqrt{3}}{4}\right)\right] \\ &= \frac{7}{6}\pi - \frac{\sqrt{3}}{2}. \end{aligned}$

Thus, the correct answer is E.

2003 AMC 12B Problem 16

Answer: E

Difficulty rating: 1680

Solution:

Problem 16 in Other Years