## 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