AMC10 2004 B

AMC10 2004 B · Q9

AMC10 2004 B · Q9. It mainly tests Circle theorems, Area & perimeter.

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?

一个边长为 10 的正方形，其一顶点为中心半径为 10 的圆。该正方形与圆包围区域的并集面积是多少？

(A) 200 + 25π 200 + 25π

(B) 100 + 75π 100 + 75π

(D) 100 + 100π 100 + 100π

(E) 100 + 125π 100 + 125π

Answer

Correct choice: (B)

正确答案：（B）

Solution

The areas of the regions enclosed by the square and the circle are $10^2 = 100$ and $\pi(10)^2 = 100\pi$, respectively. One quarter of the second region is also included in the first, so the area of the union is $100 + 100\pi - 25\pi = 100 + 75\pi$.

正方形与圆包围区域的面积分别为 $10^2 = 100$ 和 $\pi(10)^2 = 100\pi$。第二个区域的四分之一也包含在第一个区域中，因此并集面积为 $100 + 100\pi - 25\pi = 100 + 75\pi$。

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