Question

下图显示了 $13$ 个半径为 $1$ 的圆位于一个更大的圆内。所有交点都发生在相切点处。图中阴影区域，也就是在大圆内但在所有半径为 $1$ 的圆外的区域，面积是多少？

The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1?$

$4\pi\sqrt{3}$

$7\pi$

$\pi(3\sqrt{3} + 2)$

$10\pi(\sqrt{3} - 1)$

$\pi(\sqrt{3} + 6)$

答案：A

知识点：圆面积相切圆面积分割

难度评级：1500

解答：

在中心放一个单位圆，六个圆围绕它且圆心距中心为 $2$ （形成一个六边形），再在外侧空隙中放六个圆，其圆心距中心为 $2\sqrt{3}$ 。这共有 $1 + 6 + 6 = 13$ 个圆。

最外层圆与大圆相切，所以大圆半径为 $2\sqrt{3} + 1$ 。它的面积为 $\pi(2\sqrt{3} + 1)^2 = \pi(13 + 4\sqrt{3}).$

减去 $13$ 个单位圆的面积，剩下 $\pi(13 + 4\sqrt{3}) - 13\pi = 4\pi\sqrt{3}$ 。

所以正确答案是 A。

Place a unit circle at the center, six around it with centers at distance $2$ (a hexagon), and six more with centers at distance $2\sqrt{3}$ in the outer gaps. That is $1 + 6 + 6 = 13$ circles.

The outermost circles are tangent to the big circle, whose radius is therefore $2\sqrt{3} + 1.$ Its area is $\pi(2\sqrt{3} + 1)^2 = \pi(13 + 4\sqrt{3}).$

Subtracting the $13$ unit circles leaves $\pi(13 + 4\sqrt{3}) - 13\pi = 4\pi\sqrt{3}.$

Thus, the correct answer is A.

2019 AMC 12A 第 10 题

答案：A

难度评级：1500

解答：

其他年份的第 10 题