AMC12 2004 B

AMC12 2004 B · Q7

AMC12 2004 B · Q7. It mainly tests Area & perimeter, Geometry misc.

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?

一个正方形的边长为 $10$，并且以它的一个顶点为圆心作半径为 $10$ 的圆。由正方形和圆所围成区域的并集面积是多少？

(A) 200 + 25\pi 200 + 25\pi

(B) 100 + 75\pi 100 + 75\pi

(D) 100 + 100\pi 100 + 100\pi

(E) 100 + 125\pi 100 + 125\pi

Answer

Correct choice: (B)

正确答案：（B）

Solution

The area of the circle is $S_{\bigcirc}=100\pi$; the area of the square is $S_{\square}=100$. Exactly $\frac{1}{4}$ of the circle lies inside the square. Thus the total area is $\dfrac34 S_{\bigcirc}+S_{\square}=\boxed{\mathrm{(B)\ }100+75\pi}$.

圆的面积为 $S_{\bigcirc}=100\pi$；正方形的面积为 $S_{\square}=100$。圆的恰好 $\frac{1}{4}$ 位于正方形内部。因此总面积为 $\dfrac34 S_{\bigcirc}+S_{\square}=\boxed{\mathrm{(B)\ }100+75\pi}$。

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