2002 AMC 10B Problem 5

Problem 5 of 25EasierGeometry

Circles of radius

2

and

3

are externally tangent and are circumscribed by a third circle, as shown in the figure. What is the area of the shaded region?

Show solution

Solution

The two small circles line up along a diameter of the big circle, so that diameter is

2\cdot3 + 2\cdot2 = 10

and the large radius is

5.

The shaded region is the large disk with the two small disks removed:

\begin{aligned} &\pi(5^2) - \pi(3^2) - \pi(2^2) \\ &= \pi(25 - 9 - 4) \\ &= 12\pi. \end{aligned}

Thus, the correct answer is E.