AMC10 2019 A

AMC10 2019 A · Q16

AMC10 2019 A · Q16. It mainly tests Circle theorems, Area & perimeter.

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 of the circles of radius 1?

下图显示了一个大圆内有13个半径为1的小圆。所有相交点均为切点。图中阴影区域是大圆内部但所有半径为1的小圆外部的区域面积是多少？

(A) $4\pi\sqrt{3}$ $4\pi\sqrt{3}$

(B) $7\pi$ $7\pi$

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

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

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): Let $A, B, C,$ and $D$ be the centers of four of the circles as shown below, and let $P$ be the intersection of the diagonals of rhombus $ABDC$. Then $PC=1$ and $AC=2$, so $AP=\sqrt{3}$; similarly $PD=\sqrt{3}$. The radius of the large circle is therefore $1+2\sqrt{3}$. The requested area is \[ \pi(1+2\sqrt{3})^2-13\pi=4\pi\sqrt{3}. \] Note: This problem is related to the question of how densely the plane can be packed with congruent circles—how much wasted space there is with the most efficient packing. It has been proved that the best arrangement is the one shown in this problem, with each circle surrounded by six others. The fraction of the plane covered by the circles is $\frac{\pi}{6}\sqrt{3}\approx0.9069$.

答案（A）：设 $A, B, C, D$ 为如下图所示四个圆的圆心，设 $P$ 为菱形 $ABDC$ 的两条对角线的交点。则 $PC=1$ 且 $AC=2$，所以 $AP=\sqrt{3}$；同理 $PD=\sqrt{3}$。因此大圆的半径为 $1+2\sqrt{3}$。所求面积为 \[ \pi(1+2\sqrt{3})^2-13\pi=4\pi\sqrt{3}. \] 注：本题与“平面如何用全等圆最密堆积”有关——即在最有效的堆积中有多少空间被浪费。已证明最优排列就是本题所示的排列，其中每个圆被另外六个圆包围。平面被圆覆盖的比例为 $\frac{\pi}{6}\sqrt{3}\approx0.9069$。

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