2003 AMC 10B Problem 19

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:circle area sector equilateral triangle area decomposition

Difficulty rating: 1630

Small hint:

Start with the large semicircle’s area

\tfrac12 \pi (2)^2 = 2\pi

Big hint:

The overlaps of adjacent small semicircles are

60^\circ

sectors plus equilateral triangles

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

\begin{gathered} 2\pi - 5 \cdot \dfrac{\pi}{6} - 2 \cdot \dfrac{\sqrt3}{4} \\ = \dfrac{7}{6}\pi - \dfrac{\sqrt3}{2}. \end{gathered}

Thus, the correct answer is E.