Question

The exterior angle of a regular polygon measures 10 degrees. How many sides does the polygon have?

Answer 1

The exterior and interior angles of a polygon are supplementary angles (they add up to 180 degrees). Since the exterior angle of the polygon has a measure of 10 degrees, then the interior angles must have a measure of 170 degrees.

On the other hand, the measure of the internal angle of an n-sided polygon is:

`180*(n-2)/(n)`

Since the internal angle must have a measure of 170 degrees, then:

`180*(n-2)/(n)=170`

Solve for n:

`\begin{gathered} \Rightarrow180(n-2)=170n \\ \Rightarrow180n-360=170n \\ \Rightarrow180n-170n=360 \\ \Rightarrow10n=360 \\ \Rightarrow n=(360)/(10) \\ \Rightarrow n=36 \end{gathered}`

Therefore, the polygon has 36 sides.