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?

$200 + 25\pi$

$100 + 75\pi$

$75 + 100\pi$

$100 + 100\pi$

$100 + 125\pi$

答案：B

知识点：圆面积面积分割

难度评级：1220

解答：

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.

2004 AMC 12B 第 7 题

答案：B

难度评级：1220

解答：

其他年份的第 7 题