2008 AMC 10A Problem 17

Question

An equilateral triangle has side length

6.

What is the area of the region containing all points that are outside the triangle and not more than

3

units from a point of the triangle?

36 + 24\sqrt{3}

54 + 9\pi

54 + 18\sqrt{3} + 6\pi

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

9\left(\sqrt{3} + 1\right)^2 \pi

The region is three rectangles along the sides plus three sectors at the corners

The three corner sectors are each

120^\circ,

and together they make one full circle of radius

3

Along each of the three sides is a

6 \times 3

rectangle, contributing

3 \cdot 6 \cdot 3 = 54.

At each vertex is a

120^\circ

sector of radius

3;

the three together form a full circle of area

\pi \cdot 3^2 = 9\pi.

The total area is

54 + 9\pi.

Thus, the correct answer is B.

Problem 17 in other years