AMC8 2018

AMC8 2018 · Q15

AMC8 2018 · Q15. It mainly tests Ratios & proportions, Area & perimeter.

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of 1 square unit, then what is the area of the shaded region, in square units?

下图中，两个小圆的直径均为大圆的半径。如果两个小圆的总面积为1平方单位，则阴影区域的面积是多少平方单位？

(A) $\frac{1}{4}$ $\frac{1}{4}$

(B) $\frac{1}{3}$ $\frac{1}{3}$

(D) 1 1

(E) $\frac{\pi}{2}$ $\frac{\pi}{2}$

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): If one of the smaller circles has radius $r$, then its area is $\pi r^2$, and the area of the larger circle is $\pi(2r)^2=4\pi r^2$. Thus the area of the large circle is $4$ times the area of one of the small circles. So the area of the shaded region is equal to the combined area of the two smaller circles, which is $1$ square unit.

答案（D）：如果较小的一个圆的半径为 $r$，那么它的面积是 $\pi r^2$，而较大圆的面积是 $\pi(2r)^2=4\pi r^2$。因此，大圆的面积是小圆面积的 $4$ 倍。所以阴影部分的面积等于两个小圆面积之和，也就是 $1$ 平方单位。

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