2009 AMC 10A 第 6 题

Question

如图所示，一个半径为

2

的圆内切于一个半圆。半圆内但圆外的部分被涂阴影。阴影部分占半圆面积的几分之几？

A circle of radius

2

is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle’s area is shaded?

\dfrac{1}{2}

\dfrac{\pi}{6}

\dfrac{2}{\pi}

\dfrac{2}{3}

\dfrac{3}{\pi}

答案：A

知识点：圆面积相切圆

难度评级：1020

小提示：

小圆放在直径上并与弧相切，所以半圆的半径为

4

。

The circle sits on the diameter and touches the arc, so the semicircle has radius

4

大提示：

比较小圆面积与半圆面积。

Compare the circle’s area to the semicircle’s area

解答：

内切圆落在直径上并与半圆弧相切，所以半圆的半径是

4

。半圆面积为

\dfrac12 \pi (4)^2 = 8\pi\text{。}

小圆面积为

\pi(2)^2 = 4\pi

，所以阴影面积为

8\pi - 4\pi = 4\pi

。

阴影部分占半圆的比例为

\dfrac{4\pi}{8\pi} = \dfrac12

。

所以正确答案是 A。

The inscribed circle rests on the diameter and is tangent to the arc, so the semicircle has radius

4.

Its area is

\dfrac12 \pi (4)^2 = 8\pi.

The circle’s area is

\pi(2)^2 = 4\pi,

so the shaded area is

8\pi - 4\pi = 4\pi.

The shaded fraction is

\dfrac{4\pi}{8\pi} = \dfrac12.

Thus, the correct answer is A.

2009 AMC 10A 第 6 题

答案：A

难度评级：1020

小提示：

大提示：

解答：

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