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?

$\pi-2$

$\dfrac{\pi}{2}$

$\dfrac{\pi\sqrt3}{3}$

$2(\pi-2)$

$\pi$

Answer: D

Concepts:sector area decomposition symmetry

Difficulty rating: 1580

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.

2007 AMC 10B Problem 13

Answer: D

Difficulty rating: 1580

Solution:

Problem 13 in Other Years