Question

三个半径为 $2$ 的圆两两相切。如图，求这三个圆以及它们围成的中间区域的总面积。

Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?

$10\pi+4\sqrt{3}$

$13\pi-\sqrt{3}$

$12\pi+\sqrt{3}$

$10\pi+9$

$13\pi$

连接三个圆心。

Connect the three circle centers

所求面积是三个 $300^\circ$ 扇形加上一个等边三角形。

The desired area is three $300^\circ$ sectors plus an equilateral triangle

连接三个圆心，得到边长为 $4$ 的等边三角形，面积为 $4\sqrt3$ 。

每个圆中计入的部分是 $300^\circ$ 扇形，面积为 $\dfrac{300}{360}\pi\cdot2^2=\dfrac{10\pi}{3}$ 。

所以总面积为 $3\cdot\dfrac{10\pi}{3}+4\sqrt3=10\pi+4\sqrt3$ 。

所以正确答案是 A。

Connect the centers of the three circles. This forms an equilateral triangle of side $4$ , with area $4\sqrt3$ .

The included part of each circle is a $300^\circ$ sector, whose area is $\dfrac{300}{360}\pi\cdot2^2=\dfrac{10\pi}{3}$ .

The total area is $3\cdot\dfrac{10\pi}{3}+4\sqrt3=10\pi+4\sqrt3$ .

Thus, A is the correct answer.