2002 AMC 12A Problem 5

Problem 5 of 25EasierGeometry

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.

Show solution

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.

Answer choices

Solution

More practice