Question

一个边长为 $2$ 的正方形和一个圆有相同的中心。圆内正方形外的区域总面积等于圆外正方形内的区域总面积。圆的半径是多少？

A square with side length $2$ and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

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

$\dfrac{1+\sqrt{2}}{2}$

$\dfrac{3}{2}$

$\sqrt{3}$

$\sqrt{\pi}$

答案：A

知识点：圆面积正方形（几何）

难度评级：1560

解答：

设 $S$ 为同时在正方形和圆内的公共面积。题目说明圆独有区域面积等于正方形独有区域面积。

在两个相等面积上都加上 $S$ ，可知圆的总面积等于正方形的总面积。

正方形面积为 $2^2=4$ ，所以 $\pi r^2=4$ 。因此 $r=\dfrac{2}{\sqrt{\pi}}$ 。

所以正确答案是 A。

Let $S$ be the common area inside both the square and circle. The problem says the circle-only area equals the square-only area.

Adding $S$ to both equal areas shows that the total area of the circle equals the total area of the square.

The square area is $2^2=4$ , so $\pi r^2=4$ . Hence $r=\dfrac{2}{\sqrt{\pi}}$ .

Thus, A is the correct answer.

2005 AMC 8 第 25 题

答案：A

难度评级：1560

解答：

其他年份的第 25 题