How many sides does a regular polygon have if the measure of an interior angle is $ 171^\circ $ ?

Hint: In order to determine the sides of the polygon whose measure of interior angle is $ 171^\circ $ , we will use the formula $ Interior{\text{ }}angles{\text{ }}of{\text{ }}a{\text{ }}regular{\text{ }}polygon = \dfrac{{180^\circ \left( n \right) - 360^\circ }}{n} $ , as the measure of the interior angle of a polygon is given. We will determine the $ n $ by substituting the value of measure of interior angle of a polygon and evaluating it.

Complete step-by-step answer:
We know that from interior angles of a polygon, if $ n $ is the number of sides of a polygon, then the formula is,
 $ Interior{\text{ }}angles{\text{ }}of{\text{ }}a{\text{ }}regular{\text{ }}polygon = \dfrac{{180^\circ \left( n \right) - 360^\circ }}{n} $
Where $ n $ is the number of sides.
It is given that the measure of interior angle of a polygon is $ 171^\circ $ .
Therefore, $ 171^\circ = \dfrac{{180^\circ \left( n \right) - 360^\circ }}{n} $
 $ 171n = 180n - 360 $
 $ 180n - 171n = 360 $
  $ 9n = 360 $
 $ n = \dfrac{{360}}{9} $
 $ n = 40 $
Hence, the sides of the polygon whose measure of interior angle is $ 171^\circ $ is $ 40 $
So, the correct answer is “ $ 40 $ ”.

Note: An interior angle of a polygon is an angle formed inside the two adjacent sides of a polygon. In other words, the angles measure at the interior part of a polygon is called the interior angle of a polygon.
We have other methods also to determine the interior angles of a polygon.
If the sum of interior angles of a polygon is given by the formula,
 $ Interior{\text{ }}angles{\text{ }}of{\text{ }}a{\text{ }}regular{\text{ }}polygon = 180^\circ \left( {n - 2} \right) $
Where $ n $ is the number of sides.
If the exterior angle of a polygon is given, then the formula to determine the interior angle is,
\[Interior{\text{ }}angle{\text{ }}of{\text{ }}a{\text{ }}polygon\; = 180^\circ - exterior{\text{ }}angle{\text{ }}of{\text{ }}a{\text{ }}polygon\]
If we know the sum of all interior angles of a regular polygon, then we can obtain the interior angle by dividing the sum by the number of sides.
\[Interior{\text{ }}angle = \dfrac{{sum{\text{ }}of{\text{ }}interior{\text{ }}angles{\text{ }}of{\text{ }}a{\text{ }}polygon}}{n}\], where $ n $ is the number of sides.