AMC12 2019 A

AMC12 2019 A · Q10

AMC12 2019 A · Q10. It mainly tests Area & perimeter, Coordinate geometry.

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?

下面的图形显示一个大圆内有 13 个半径为 1 的圆。所有交点均发生在相切点处。图形中阴影区域是大圆内部但所有半径为 1 的圆外部的区域面积是多少？

(A) 4π√3 4π√3

(B) 7π 7π

(D) 10π(√3 − 1) 10π(√3 − 1)

(E) π(√3 + 6) π(√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 $$ \pi(1 + 2\sqrt{3})^2 - 13\pi = 4\pi\sqrt{3}. $$

答案（A）：设 $A$、$B$、$C$、$D$ 为如下图所示四个圆的圆心，并令 $P$ 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 $$ \pi(1 + 2\sqrt{3})^2 - 13\pi = 4\pi\sqrt{3}. $$

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