1994 AMC 12 第 17 题

Question

一个长为

8

、宽为

2\sqrt2

的长方形与一个半径为

2

的圆有相同的中心。长方形与圆的公共区域面积是

An

8

by

2\sqrt2

rectangle has the same center as a circle of radius

2.

The area of the region common to both the rectangle and the circle is

2\pi

2\pi+2

4\pi-4

2\pi+4

4\pi-2

长方形从圆中截去了两个全等的弓形

The rectangle removes two congruent caps from the circle

弓形的弦到圆心的距离为

\sqrt2

，所以它所对圆心角的一半为

45^\circ

The cap chord is at distance

\sqrt2

from the center, so its half central angle is

45^\circ

长方形比圆宽，所以公共区域是圆去掉与圆心距离超过

\sqrt2

的上、下两个弓形。一个弓形的面积为

\begin{aligned} 2^2\left(\frac{\pi}{4}\right) &-\sqrt2\sqrt{2^2-(\sqrt2)^2}\\ &=\pi-2 \end{aligned}\text{。}

因此公共区域面积为

4\pi-2(\pi-2)=2\pi+4

。所以正确答案是 D。

The rectangle is wider than the circle, so the common region is the circle with the top and bottom caps beyond distance

\sqrt2

from the center removed. One cap has area

\begin{aligned} 2^2\left(\frac{\pi}{4}\right) &-\sqrt2\sqrt{2^2-(\sqrt2)^2}\\ &=\pi-2. \end{aligned}

Therefore the common area is

4\pi-2(\pi-2)=2\pi+4.

Thus the correct answer is D.

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