2005 AMC 10B 第 7 题

Question

一个圆内切于一个正方形，接着一个正方形内接于这个圆，最后一个圆内切于这个正方形。较小圆的面积与较大正方形的面积之比是多少？

A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?

\dfrac{\pi}{16}

\dfrac{\pi}{8}

\dfrac{3\pi}{16}

\dfrac{\pi}{4}

\dfrac{\pi}{2}

答案：B

知识点：圆面积正方形（几何）面积比

难度评级：1240

小提示：

设较小圆半径为

r

，然后向外推。

Let the smaller circle have radius

r

and work outward

大提示：

较大圆的半径是较小正方形对角线的一半，即

\sqrt2\,r

。

The larger circle’s radius is half the diagonal of the smaller square,

\sqrt2\,r

解答：

设较小圆半径为

r

，则它的面积为

\pi r^2

。

外切于这个小圆的较小正方形边长为

2r

，其对角线

2\sqrt2\,r

是较大圆的直径。所以较大圆半径为

\sqrt2\,r

。

外切于较大圆的较大正方形边长为

2\sqrt2\,r

，面积为

8r^2

。

所求比为

\dfrac{\pi r^2}{8r^2} = \dfrac{\pi}{8}\text{。}

所以正确答案是 B。

Let the smaller circle have radius

r,

so its area is

\pi r^2.

The smaller square, which circumscribes this circle, has side

2r,

and its diagonal

2\sqrt2\,r

is the diameter of the larger circle. So the larger circle has radius

\sqrt2\,r.

The larger square circumscribes the larger circle, so it has side

2\sqrt2\,r

and area

8r^2.

The desired ratio is

\dfrac{\pi r^2}{8r^2} = \dfrac{\pi}{8}.

Thus, B is the correct answer.

2005 AMC 10B 第 7 题

答案：B

难度评级：1240

小提示：

大提示：

解答：

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