Question

How many sides does a regular polygon have if each of its interior angles is \[165^\circ \]?
1) 24
2) 44
3) 55
4) 33

Answer 1

Hint: Here, we will first take the given angle of a regular polygon equal to \[\dfrac{{180^\circ \left( {n - 2} \right)}}{n}\]. Then we will simplify the obtained equation to find the value of \[n\].

Complete step-by-step answer:
We are given that each of the interior angles of a regular polygon is \[165^\circ \].

Let us assume that the number of sides of a regular polygon is represented by \[n\].

We know that the measure of each interior angle of a regular polygon the formula is given by \[\dfrac{{180^\circ \left( {n - 2} \right)}}{n}\].

Since the measure of each interior angle is \[165^\circ \], we have

\[ \Rightarrow \dfrac{{180^\circ \left( {n - 2} \right)}}{n} = 165^\circ \]

Multiplying the above equation by \[n\] on both sides, we get

\[
   \Rightarrow 180^\circ \left( {n - 2} \right) = 165^\circ n \\
   \Rightarrow 180^\circ n - 360^\circ = 165^\circ n \\
 \]

Adding the above equation with \[360^\circ \] on each side, we get

\[
   \Rightarrow 180^\circ n - 360^\circ + 360^\circ = 165^\circ n + 360^\circ \\
   \Rightarrow 180^\circ n = 165^\circ n + 360^\circ \\
 \]

Subtracting both sides by \[165^\circ n\] on both sides, we get

\[
   \Rightarrow 180^\circ n - 165^\circ n = 165^\circ n + 360^\circ - 165^\circ n \\
   \Rightarrow 15^\circ n = 360^\circ \\
 \]

Dividing the above equation by \[15^\circ \] on each side, we get

\[
   \Rightarrow \dfrac{{15^\circ n}}{{15^\circ }} = \dfrac{{360^\circ }}{{15^\circ }} \\
   \Rightarrow n = 24^\circ \\
 \]


Thus, the regular polygon with interior angle each measuring \[165^\circ \] is a 24-sided polygon.

Hence, our option A is correct.

Note: We have to always remember that the sum of all the exterior angles in any polygon is 360 degree. In solving these types of questions, we have to take the given angle of a regular polygon equal to \[\dfrac{{180^\circ \left( {n - 2} \right)}}{n}\]. Then this question will be very simple to solve and compute the value of \[n\].