MathDB
Circles in circles

Source: 2019 AMC 10A #16 / AMC 12A #10

February 8, 2019
AMCAMC 102019 AMC2019 AMC 12A2019 AMC 10AgeometryAMC 12

Problem Statement

The figure below shows 1313 circles of radius 11 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?1 ?
[asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy]
<spanclass=′latex−bold′>(A)</span>4π3<spanclass=′latex−bold′>(B)</span>7π<spanclass=′latex−bold′>(C)</span>π(33+2)<spanclass=′latex−bold′>(D)</span>10π(3−1)<spanclass=′latex−bold′>(E)</span>π(3+6)<span class='latex-bold'>(A) </span> 4 \pi \sqrt{3} \qquad<span class='latex-bold'>(B) </span> 7 \pi \qquad<span class='latex-bold'>(C) </span> \pi(3\sqrt{3} +2) \qquad<span class='latex-bold'>(D) </span> 10 \pi (\sqrt{3} - 1) \qquad<span class='latex-bold'>(E) </span> \pi(\sqrt{3} + 6)
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