2003 AMC 12B Problem 16

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

Small hint:

The large semicircle has area

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

Big hint:

Use inclusion-exclusion: subtract the two overlaps from the total area of the three small semicircles

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.