2009 AMC 10A Problem 6

Question

A circle of radius

2

is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle’s area is shaded?

\dfrac{1}{2}

\dfrac{\pi}{6}

\dfrac{2}{\pi}

\dfrac{2}{3}

\dfrac{3}{\pi}

Answer: A

Concepts:circle area tangent circles

Difficulty rating: 1020

Small hint:

The circle sits on the diameter and touches the arc, so the semicircle has radius

4

Big hint:

Compare the circle’s area to the semicircle’s area

Solution:

The inscribed circle rests on the diameter and is tangent to the arc, so the semicircle has radius

4.

Its area is

\dfrac12 \pi (4)^2 = 8\pi.

The circle’s area is

\pi(2)^2 = 4\pi,

so the shaded area is

8\pi - 4\pi = 4\pi.

The shaded fraction is

\dfrac{4\pi}{8\pi} = \dfrac12.

Thus, the correct answer is A.