AMC8 1992

AMC8 1992 · Q24

AMC8 1992 · Q24. It mainly tests Circle theorems, Area & perimeter.

Four circles of radius 3 are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to

四个半径为3的圆按所示排列。它们的圆心是正方形的顶点。阴影区域的面积最接近

(A) 7.7 7.7

(B) 12.1 12.1

(D) 18 18

(E) 27 27

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): The four quarter-circles that lie inside the square have a total area equal to the area of one of the circles, $9\pi$. The length of a side of the square is equal to two radii, $6$, and thus the square has area $36$. The difference is $36-9\pi<36-9(3)=9$, so it is closest to $7.7$. (the area, to one decimal place, is $7.7$.)

答案（A）：正方形内的四个四分之一圆的总面积等于其中一个圆的面积，即 $9\pi$。正方形的边长等于两个半径之和，为 $6$，因此正方形的面积是 $36$。两者的差为 $36-9\pi<36-9(3)=9$，所以最接近 $7.7$。（该面积保留一位小数为 $7.7$。）

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