Question

How many sides does a regular polygon have if each of its interior angles is ${165^0}$?
A.24
B.26
C.16
D.18

Answer 1

Hint: First use the supplementary condition to find an exterior polygon. Use the result of the sum of exterior angles of the regular polygon to find the number of sides.

Complete step-by-step answer:
In geometry, a polygon is a plane figure that is described by a finite number of straight-line segments connected to form a closed polygonal chain or polygonal circuit.
A regular polygon is a polygon that is equiangular and equilateral. It means that all the angles and sides of the polygon are equal respectively.
An exterior angle of a polygon is an angle formed at the vertex of a polygon exterior to it. It is formed by a side and the extension of another side adjacent to it.
Given the problem, we are given a regular polygon.
Hence all the interior and exterior angles of this polygon are equal respectively.
Given each of its interior angles to be ${165^0}$.
We know that if the interior angle of the polygon is ${\theta ^0}$, the exterior angle is given by ${180^0} - {\theta ^0}$
Hence each exterior angle of the given polygon is equal to ${180^0} - {165^0} = {15^0}$.
Also, the sum of exterior angles of a polygon is equal to ${360^0}$.
Also, the number of exterior angles of a polygon is equal to the number of sides of a polygon.
Hence the number of sides of the polygon will be given by,
${\text{Sides of a regular polygon}} = \dfrac{{{\text{36}}{{\text{0}}^0}}}{{{\text{Measure of the exterior angle}}}}$

$ \Rightarrow {\text{Sides of a regular polygon}} = \dfrac{{{\text{36}}{{\text{0}}^0}}}{{{{15}^0}}} = 24$
Hence the number of sides of the regular polygon is equal to 24.

Note: The geometrical properties of the regular polygon should be kept in mind while solving problems like above. An irregular polygon does not have all sides equal and all angles equal. A polygon can also be differentiated as convex and concave. If each of the interior angles of a polygon is less than 180 degrees, then it is called convex polygon. If at least one angle of a polygon is more than 180 degree, then it is called a concave polygon. A regular polygon is a type of convex polygon.