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

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

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

解答：

其他年份的第 6 题