Question

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 $1?$

$4\pi\sqrt{3}$

$7\pi$

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

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

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

Solution:

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 Problem 10

Solution:

Problem 10 in Other Years