Question

在半径为 $2$ 的半圆的直径 $\overline{AB}$ 上构造三个半径为 $1$ 的半圆。小半圆的圆心将 $\overline{AB}$ 分成四段相等的线段，如图所示。位于大半圆内部且在较小半圆外部的阴影区域面积是多少？

Three semicircles of radius $1$ are constructed on diameter $\overline{AB}$ of a semicircle of radius $2.$ The centers of the small semicircles divide $\overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?

$\pi - \sqrt{3}$

$\pi - \sqrt{2}$

$\dfrac{\pi + \sqrt{2}}{2}$

$\dfrac{\pi + \sqrt{3}}{2}$

$\dfrac{7}{6}\pi - \dfrac{\sqrt{3}}{2}$

答案：E

知识点：圆面积扇形等边三角形面积分割

难度评级：1630

解答：

大半圆面积为 $\dfrac12 \pi (2)^2 = 2\pi$ 。

删去的部分等于五个半径为 $1$ 的 $60^\circ$ 扇形和两个边长为 $1$ 的等边三角形；每个扇形面积为 $\dfrac{\pi}{6}$ ，每个等边三角形面积为 $\dfrac{\sqrt3}{4}$ 。

阴影面积为 $\begin{gathered} 2\pi - 5 \cdot \dfrac{\pi}{6} - 2 \cdot \dfrac{\sqrt3}{4} \\ = \dfrac{7}{6}\pi - \dfrac{\sqrt3}{2}. \end{gathered}$

所以正确答案是 E。

The large semicircle has area $\dfrac12 \pi (2)^2 = 2\pi.$

Removing the small semicircles deletes a region equal to five congruent $60^\circ$ sectors of radius $1$ plus two equilateral triangles of side $1.$ Each sector has area $\dfrac{\pi}{6}$ and each triangle has area $\dfrac{\sqrt3}{4}.$

The shaded area is $\begin{gathered} 2\pi - 5 \cdot \dfrac{\pi}{6} - 2 \cdot \dfrac{\sqrt3}{4} \\ = \dfrac{7}{6}\pi - \dfrac{\sqrt3}{2}. \end{gathered}$

Thus, the correct answer is E.

2003 AMC 10B 第 19 题

答案：E

难度评级：1630

解答：

其他年份的第 19 题