Question

半径为 $2$ 和 $3$ 的两个圆外切，并被第三个圆外接，如图所示。阴影区域的面积是多少？

Circles of radius $2$ and $3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. What is the area of the shaded region?

$3\pi$

$4\pi$

$6\pi$

$9\pi$

$12\pi$

答案：E

知识点：相切圆圆面积

难度评级：1020

小提示：

两个小圆沿着大圆的一条直径排列。

The two small circles lie along a diameter of the large circle

大提示：

阴影面积等于大圆面积减去两个小圆面积。

The shaded area is the large disk minus the two smaller disks

解答：

两个小圆沿大圆的一条直径排列，所以大圆直径为 $2\cdot3 + 2\cdot2 = 10$ ，半径为 $5$ 。

阴影面积为 $\begin{aligned} &\pi(5^2) - \pi(3^2) - \pi(2^2) \\ &= \pi(25 - 9 - 4) \\ &= 12\pi \end{aligned}\text{。}$

所以正确答案是 E。

The two small circles line up along a diameter of the big circle, so that diameter is $2\cdot3 + 2\cdot2 = 10$ and the large radius is $5.$

The shaded region is the large disk with the two small disks removed: $\begin{aligned} &\pi(5^2) - \pi(3^2) - \pi(2^2) \\ &= \pi(25 - 9 - 4) \\ &= 12\pi. \end{aligned}$

Thus, the correct answer is E.

2002 AMC 10B 第 5 题

答案：E

难度评级：1020

小提示：

大提示：

解答：

其他年份的第 5 题