2002 AMC 10A Problem 19

Question

Spot’s doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside the doghouse that Spot can reach?

\dfrac{2}{3}\pi

2\pi

\dfrac{5}{2}\pi

\dfrac{8}{3}\pi

3\pi

Answer: E

Concepts:sector regular polygon circle area

Difficulty rating: 1600

Small hint:

The interior angle of a regular hexagon is

120^\circ,

so Spot sweeps

240^\circ

at radius

2

Big hint:

Around each adjacent vertex the leftover

1

yard of rope sweeps a

60^\circ

sector

Solution:

At the tether vertex the hexagon blocks its

120^\circ

interior angle, leaving a

240^\circ

sector of radius

2:

area

\dfrac{240}{360}\pi(2)^2=\dfrac{8\pi}{3}.

Wrapping around each of the two adjacent vertices,

1

yard of rope remains and sweeps a

60^\circ

sector:

2\cdot\dfrac{60}{360}\pi(1)^2=\dfrac{\pi}{3}.

The total is

\dfrac{8\pi}{3}+\dfrac{\pi}{3}=3\pi.

Thus, the correct answer is E.