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2009 AMC 10A Problem 6

2009 AMC 10A Problem 6AMC 10 Step by Step

Name: Date: September 13, 2026

Circles · difficulty 2 of 5 · about 2 min · position P1-10

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?

A)
$\frac{1}{2}$
B)
$\frac{\pi}{6}$
C)
$\frac{2}{\pi}$
D)
$\frac{2}{3}$
E)
$\frac{3}{\pi}$

Answer

Key insight

The inscribed circle's diameter equals the semicircle's radius, so the semicircle has radius 4 and area 8 pi, exactly twice the circle's 4 pi.

Solution

The small circle touches the semicircle's diameter at the center of the semicircle and touches the arc at the top. So the small circle's diameter, $4$ , runs from the center of the semicircle straight up to the arc; that distance is the semicircle's radius. Hence the semicircle has radius $4$ .

Semicircle area: $\frac{1}{2} \pi \cdot 4^2 = 8\pi$ .

Circle area: $\pi \cdot 2^2 = 4\pi$ .

Shaded area: $8\pi - 4\pi = 4\pi$ , which is $\dfrac{4\pi}{8\pi} = \dfrac{1}{2}$ of the semicircle.

The answer is $\boxed{\textbf{(A)}\ \frac{1}{2}}$ .

Why this works

The only real step is reading the figure correctly: a circle inscribed in a semicircle has its diameter along the semicircle's radius. After that it is two area formulas and a subtraction. Whenever a figure shows nested circles, locate the segment that both shapes share (here, center to top of arc) and express it two ways.

The trap

Giving the semicircle radius 2 as well, or using a full circle of radius 4 (area 16 pi) and answering 1/4 or 3/4.

Common mistakes

Giving the semicircle radius 2 as well, or using a full circle of radius 4 (area 16 pi) and answering 1/4 or 3/4.
Reporting the shaded area $4\pi$ or the ratio of circle to semicircle instead of the requested fraction.

Techniques

Set up the equation/formula and compute; no special trick needed