2002 AMC 10A 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:tangent circles circle area

Difficulty rating: 1060

Small hint:

The large radius spans the center circle plus two more radii

Big hint:

Subtract the seven unit circles from the large circle

Solution:

The center of a surrounding circle is

2

from the center (two radii), and adding its own radius

1

gives a large radius of

3.

The large circle has area

9\pi,

and the seven unit circles have total area

7\pi,

so the shaded region is

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

Thus, the correct answer is C.