2002 AMC 12A Problem 5

Question

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

\pi

1.5\pi

2\pi

3\pi

3.5\pi

Answer: C

Concepts:circle area tangent circles

Difficulty rating: 1270

Small hint:

Each surrounding circle has its center

2

units from the center, so the large circle has radius

3

Big hint:

Subtract the areas of the seven unit circles from the area of the large circle

Solution:

Each of the six outer unit circles is tangent to the central unit circle, so its center is

2

units from the center. Adding one more radius, the large circle has radius

3

and area

9\pi.

The seven unit circles have total area

7\pi,

so the shaded region has area

9\pi - 7\pi = 2\pi.

Thus, the correct answer is C.