2019 AMC 12A Problem 10

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)

Answer: A

Concepts:circle area tangent circles area decomposition

Difficulty rating: 1500

Small hint:

A central circle is ringed by six circles, then six more sit in the outer notches

Big hint:

The outermost centers are at distance

2\sqrt{3}

from the center, so the big radius is

2\sqrt{3} + 1

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.