2012 AMC 10B Problem 16

Question

Three circles with radius

2

are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?

10\pi+4\sqrt{3}

13\pi-\sqrt{3}

12\pi+\sqrt{3}

10\pi+9

13\pi

Connect the three circle centers

The desired area is three

300^\circ

sectors plus an equilateral triangle

Connect the centers of the three circles. This forms an equilateral triangle of side

4

, with area

4\sqrt3

.

The included part of each circle is a

300^\circ

sector, whose area is

\dfrac{300}{360}\pi\cdot2^2=\dfrac{10\pi}{3}

.

The total area is

3\cdot\dfrac{10\pi}{3}+4\sqrt3=10\pi+4\sqrt3

.

Thus, A is the correct answer.

Problem 16 in other years