2010 AMC 8 第 23 题

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 of circle

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

小提示：

两个半圆合起来的面积等于一个半径为

1

的圆的面积

The two semicircles together have the area of one circle of radius

1

.

大提示：

从坐标求圆

O

的半径

Find the radius of circle

O

from the coordinates.

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

小提示：

大提示：

解答：

其他年份的第 23 题