2003 AMC 12B 第 16 题

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

小提示：

大半圆面积为

\dfrac{1}{2}\pi(2)^2 = 2\pi

The large semicircle has area

\dfrac{1}{2}\pi(2)^2 = 2\pi

大提示：

用容斥原理：从三个小半圆的总面积中减去两块重叠部分

Use inclusion-exclusion: subtract the two overlaps from the total area of the three small semicircles

解答：

大半圆面积为

\dfrac{1}{2}\pi(2)^2 = 2\pi

。

在计入重叠之前，三个小半圆的总面积为

\dfrac{3\pi}{2}

。每相邻两个小半圆的圆心相距

1

，所以它们的公共部分由两个

60^\circ

扇形和一个等边三角形围成。因此两块重叠部分的面积各为

\dfrac{\pi}{3} - \dfrac{\sqrt{3}}{4}

。

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}\text{。}

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

小提示：

大提示：

解答：

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