Spot’s Doghouse

Spot’s doghouse has a regular hexagonal base that measures one yard on each side. Spot 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?

Source: NCTM Mathematics Teacher

SOLUTION

Spot can go anywhere in the two 120^\circ sectors of radius 2 and in the two 60^\circ sectors of radius 1.
Area of the two 120^\circ sectors
\displaystyle\frac{240^\circ}{360^\circ}\pi 2^2=\frac{8\pi}{3}
Area of the two 60^\circ sectors
\displaystyle\frac{120^\circ}{360^\circ}\pi 1^2=\frac{\pi}{3}
Total area outside the doghouse that Spot can reach
\displaystyle\frac{8\pi}{3}+\frac{\pi}{3}=3\pi

Answer: 3\pi square yards

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About mvtrinh

Retired high school math teacher.
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