2004 AMC 10B 第 9 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

将正方形面积和圆面积相加，再减去重叠部分。

Add the areas of the square and the circle, then subtract the overlap

重叠部分是圆中位于正方形内部的四分之一圆。

The overlap is the quarter of the circle that lies inside the square

正方形面积为

10^2 = 100

，圆面积为

\pi(10)^2 = 100\pi

。

因为圆心在正方形顶点，正好有四分之一圆位于正方形内部，面积为

25\pi

。

并集面积为

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

。

所以正确答案是 B。

The square has area

10^2 = 100

and the circle has area

\pi(10)^2 = 100\pi.

Since the circle is centered at a vertex of the square, exactly one quarter of the circle, area

25\pi,

lies inside the square.

The union has area

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

Thus, the correct answer is B.

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