Question

半圆 $POQ$ 和 $ROS$ 都经过圆心 $O$ 。两个半圆面积之和与圆 $O$ 面积之比是多少？

Semicircles $POQ$ and $ROS$ pass through the center $O.$ What is the ratio of the combined areas of the two semicircles to the area of circle $O?$

$\dfrac{\sqrt 2}{4}$

$\dfrac{1}{2}$

$\dfrac{2}{\pi}$

$\dfrac{2}{3}$

$\dfrac{\sqrt 2}{2}$

答案：B

知识点：圆面积勾股定理面积比

难度评级：1540

解答：

每个半圆的面积是 $\pi \dfrac {r^2}{2}$ 。它们的半径都是 $1$ ，所以两个半圆的总面积为 $\pi \frac{1}{2} + \pi \frac 12 = \pi$ 。

大圆半径等于 $OQ$ 的长度，即 $\sqrt{1^2+1^2} = \sqrt 2$ 。它的面积为 $\pi r^2 = \pi(\sqrt{2})^2 = 2\pi$ 。

因此比值为 $\dfrac{\pi}{2\pi} = \frac 12$ 。

所以正确答案是 B。

The area of each of the semicircles is $\pi \dfrac {r^2}{2} .$ Each of them has a radius of $1,$ so their combined area is $\pi \frac{1}{2} + \pi \frac 12 = \pi.$

Next, the radius of the larger circle is equal to the length of $OQ,$ which is equal to $\sqrt{1^2+1^2} = \sqrt 2.$ Its area is $\pi r^2 = \pi(\sqrt{2})^2 = 2\pi.$

This means the ratio is $\dfrac{\pi}{2\pi} = \frac 12.$

Thus, the answer is B .

2010 AMC 8 第 23 题

答案：B

难度评级：1540

解答：

其他年份的第 23 题