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?

$200 + 25\pi$

$100 + 75\pi$

$75 + 100\pi$

$100 + 100\pi$

$100 + 125\pi$

答案：B

知识点：圆面积扇形容斥原理

难度评级：1270

解答：

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.

2004 AMC 10B 第 9 题

答案：B

难度评级：1270

解答：

其他年份的第 9 题