2004 AMC 12B 第 7 题

Question

一个正方形边长为

10

，一个圆以该正方形的一个顶点为圆心，半径为

10

。正方形和圆所围成区域的并集面积是多少？

A square has sides of length

10,

and a circle centered at one of its vertices has radius

10.

What is the area of the union of the regions enclosed by the square and the circle?

200 + 25\pi

100 + 75\pi

75 + 100\pi

100 + 100\pi

100 + 125\pi

正方形与圆的重叠部分是圆的四分之一。

The overlap of the square and circle is a quarter of the circle

并集面积

= [\text{正方形}]

+ [\text{圆}]

- [\text{四分之一圆}]

。

Union

= [\text{square}]

+ [\text{circle}]

- [\text{quarter circle}]

正方形面积为

100

，圆面积为

100\pi

。二者重叠部分是位于正方形内的四分之一圆，面积为

25\pi

。并集面积为

100 + 100\pi - 25\pi = 100 + 75\pi

。

因此正确答案是 B。

The square has area

100

and the circle has area

100\pi.

Their overlap is the quarter of the circle lying inside the square, with area

25\pi.

The union is

100 + 100\pi - 25\pi = 100 + 75\pi.

Thus, the correct answer is B.

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