Question

如下图，在两个较小圆中，每个较小圆的一条直径都是大圆的一条半径。如果两个较小圆的总面积为 $1$ 平方单位，那么阴影区域的面积是多少平方单位？

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?

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

$1$

$\dfrac{\pi}{2}$

答案：D

知识点：圆面积长度、面积与体积的缩放关系

难度评级：1070

视频讲解：

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文字解答：

设大圆面积为 $A$ 。

每个小圆的直径等于大圆半径，所以每个小圆的半径是大圆半径的一半。

用符号表示，若大圆半径为 $r_l$ ，每个小圆半径为 $r_s$ ，则因为大圆面积为 $A=\pi r_l^2$ ，所以每个小圆面积为两个小圆总面积为一，于是 $2\cdot \dfrac14 A=1$ ，所以 $A=2$ 。 $r_s = \dfrac12 r_l$ $\begin{align*}\pi r_s^2 &= \pi \left(\dfrac12 r_l\right)^2 \\&= \dfrac14 (\pi r_l^2)\\&=\dfrac14 A.\end{align*}$

阴影面积等于大圆面积 $(A)$ 减去两个小圆的总面积 $(1)$ ，所以阴影面积为 $A - 1=2-1=1$ 平方单位。

所以正确答案是 D。

Let $A$ be the area of the large circle.

Since the diameter of each of the two smaller circles is itself the radius of the larger circle, the radius of each smaller circle is half that of the larger circle.

Symbolically, if we allow $r_l$ to be the radius of the large circle and $r_s$ to be the radius of each of the smaller circles: $r_s = \dfrac12 r_l$ As the area of the larger circle is equal to $A=\pi r_l^2,$ the area of the smaller circles are equal to $\begin{align*}\pi r_s^2 &= \pi \left(\dfrac12 r_l\right)^2 \\&= \dfrac14 (\pi r_l^2)\\&=\dfrac14 A.\end{align*}$ As the area of two of these smaller circles combined is equal to 1 square unit, then it follows that $2\cdot \dfrac14 A=1$ square unit, implying that $A=2$ square units.

As the area of the shaded region is equal to the area of the larger circle $(A)$ minus the combined area of the two smaller circles $(1),$ the area of the shaded region is $A - 1=2-1=1$ square unit.

Thus, the correct answer is D

2018 AMC 8 第 15 题

答案：D

难度评级：1070

视频讲解：

文字解答：

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