2003 AMC 10B 第 19 题

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

小提示：

从大半圆面积

\tfrac12 \pi (2)^2 = 2\pi

开始。

Start with the large semicircle’s area

\tfrac12 \pi (2)^2 = 2\pi

大提示：

相邻小半圆的重叠部分涉及

60^\circ

扇形和等边三角形。

The overlaps of adjacent small semicircles are

60^\circ

sectors plus equilateral triangles

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}\text{。}

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

小提示：

大提示：

解答：

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