2007 AMC 10B 第 13 题

Question

两个半径为

2

的圆分别以

(2,0)

和

(0,2)

为圆心。这两个圆内部交集的面积是多少？

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

找出两个圆的交点。

Find the two points where the circles intersect

交集的一半是一个四分之一圆减去一个直角边为

2

的等腰直角三角形。

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

2

两个圆交于

(0,0)

和

(2,2)

。

由对称性，交集的一半是从一个圆的四分之一圆中去掉直角边为

2

的等腰直角三角形。四分之一圆面积为

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

，三角形面积为

\dfrac12(2)^2=2

。

所以交集面积为

2(\pi-2)

。

所以正确答案是 D。

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.

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