2007 AMC 10B Problem 13

Question

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?

\pi-2

\dfrac{\pi}{2}

\dfrac{\pi\sqrt3}{3}

2(\pi-2)

\pi

Find the two points where the circles intersect

Half the region is a quarter-circle minus an isosceles right triangle of leg

2

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.

Problem 13 in other years