2007 AMC 10B Problem 13

Two circles of radius

2

are centered at

(2,0)

and at

(0,2).

What is the area of the intersection of the interiors of the two circles?

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Solution

The two circles intersect at

(0,0)

and

(2,2).

By symmetry, half the intersection is formed by removing an isosceles right triangle of leg length

2

from a quarter of one circle. The quarter-circle has area

\dfrac14\pi(2)^2=\pi

and the triangle has area

\dfrac12(2)^2=2.

Therefore the whole region has area

2(\pi-2).

Thus, the correct answer is D.

Tagged: sector · area decomposition · symmetry