Problem — Number of sides from the exterior angle

Text

In an equiangular polygon each exterior angle is as wide as $\dfrac{1}{3}$ of a right angle. Determine the number of sides of the polygon.

Solution

One third of a right angle measures $\frac{1}{3}\cdot 90^\circ = 30^\circ,$ so each exterior angle of the polygon measures $30^\circ$.

In an equiangular polygon the sum of the exterior angles, one per vertex, is always $360^\circ$, and the $n$ exterior angles are all congruent. Letting $n$ be the number of sides: $n \cdot 30^\circ = 360^\circ \quad\Longrightarrow\quad n = \frac{360^\circ}{30^\circ} = 12.$ The polygon therefore has $12$ sides (it is a dodecagon). $\boxed{n = 12}$