2002 AMC 10B Problem 5

Below is the professionally curated solution for Problem 5 of the 2002 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 10B solutions, or check the answer key.

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Concepts:tangent circles circle area

Difficulty rating: 1020

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?

$3\pi$

$4\pi$

$6\pi$

$9\pi$

$12\pi$

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: $\pi(5^2) - \pi(3^2) - \pi(2^2) = \pi(25 - 9 - 4) = 12\pi.$

Thus, the correct answer is E.

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