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

$2\pi$

$2\pi+2$

$4\pi-4$

$2\pi+4$

$4\pi-2$

答案：D

知识点：circle segments 面积对称性

难度评级：1960

小提示：

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

大提示：

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 $45^\circ$

解答：

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.

1994 AMC 12 第 17 题

答案：D

难度评级：1960

小提示：

大提示：

解答：

其他年份的第 17 题