Question

在半径为 $2$ 的半圆的直径 $\overline{AB}$ 上作三个半径为 $1$ 的半圆。小半圆的圆心把 $\overline{AB}$ 分成四段相等的线段，如图所示。位于大半圆内且在小半圆外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：扇形等边三角形面积分割

难度评级：1680

解答：

大半圆面积为 $\dfrac{1}{2}\pi(2)^2 = 2\pi$ 。

小半圆重叠处，相邻半圆在到两个圆心距离都为 $1$ 的点相交，形成等边三角形。从大半圆中去掉的区域由五个全等的半径为 $\dfrac{3\pi}{2}$ 的 $60^\circ$ 扇形和两个边长为 $\dfrac{\pi}{3} - \dfrac{\sqrt{3}}{4}$ 的等边三角形组成；每个扇形面积为，每个三角形面积为。

阴影面积为 $\begin{aligned} &2\pi - \left[\frac{3\pi}{2} - 2\left(\frac{\pi}{3} - \frac{\sqrt{3}}{4}\right)\right] \\ &= \frac{7}{6}\pi - \frac{\sqrt{3}}{2}. \end{aligned}$

因此，正确答案是 E。

The large semicircle has area $\dfrac{1}{2}\pi(2)^2 = 2\pi.$

The three small semicircles have total area $\dfrac{3\pi}{2}$ before their overlaps are accounted for. The centers of each adjacent pair are $1$ unit apart, so their intersection is bounded by two $60^\circ$ sectors and an equilateral triangle. Each of the two overlaps therefore has area $\dfrac{\pi}{3} - \dfrac{\sqrt{3}}{4}.$

The shaded area is $\begin{aligned} &2\pi - \left[\frac{3\pi}{2} - 2\left(\frac{\pi}{3} - \frac{\sqrt{3}}{4}\right)\right] \\ &= \frac{7}{6}\pi - \frac{\sqrt{3}}{2}. \end{aligned}$

Thus, the correct answer is E.

2003 AMC 12B 第 16 题

答案：E

难度评级：1680

解答：

其他年份的第 16 题