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 $1?$

$4 \pi \sqrt{3}$

$7 \pi$

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

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

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

Solution:

We know $\triangle ABC$ and $\triangle ABO$ are equilateral triangles.

We get that $OC = 2\sqrt{3}$ using special right triangles to find the altitudes of the triangles.

The radius of the larger circle is therefore $2\sqrt{3} + 1,$ since there is the extra unit radius after $\overline{OC}.$

The area of the larger circle is $(2\sqrt{3} + 1)^2\pi = (13 + 4\sqrt{3})\pi.$

The area of all the inner circles is $13\pi.$

The area of the shaded region is $(13 + 4\sqrt{3})\pi - 13\pi = 4\pi\sqrt{3}.$

Thus, A is the correct answer.

2019 AMC 10A Problem 16

Solution:

Problem 16 in Other Years