Question

How many sides does a regular polygon have if each of its interior angles is $165^\circ$?
A. Number of sides $= 24$
B. Number of sides $= 30$
C. Number of sides $= 36$
D. Number of sides $= 42$

Answer 1

Hint: To solve such questions start by assuming the number of sides of the given regular polygon to $n$. Next, use the formula $\dfrac{{180^\circ \left( {n - 2} \right)}}{n}$ which is the formula used to find the interior angle of a regular polygon. Then equate this formula to the given value of the interior angle.

Complete step-by-step solution:
Given the interior angle of a regular polygon is $165^\circ$.
Suppose that the number of sides of the given regular polygon is $n$.
It is known that the formula used to find the interior angle of a regular polygon is given by $\dfrac{{180^\circ \left( {n - 2} \right)}}{n}$.
As it is given that each interior angle of a regular polygon is $165^\circ$, it can be written as,
${165^\circ } = \dfrac{{180^\circ \left( {n - 2} \right)}}{n}$
Taking the denominator of RHS to the LHS, we get,
${165^\circ }n = 180^\circ \left( {n - 2} \right)$
${165^\circ }n = {180^\circ }n - {360^\circ }$
Rearranging the terms we get,
${165^\circ }n - {180^\circ }n = - {360^\circ }$
$- {15^\circ }n = - {360^\circ }$
Cancel out the minus sign from both sides, that is,
${15^\circ }n = {360^\circ }$
Divide both the sides of the equation with ${15^\circ }$ we get
$\dfrac{{{{15}^\circ }n}}{{{{15}^\circ }}} = \dfrac{{{{360}^\circ }}}{{{{15}^\circ }}}$
Canceling out the common terms we get,
$n = 24$
Hence there are $\;24$ sides for a regular polygon with each of its interior angles equal to $165^\circ$.

Therefore, option A is correct, that is, the number of sides $= 24$.

Additional Information: A diagonal is a line segment connecting two non-consecutive vertices of a polygon. A parallelogram is a quadrilateral whose opposite sides are parallel. The opposite sides of a parallelogram are of equal length.

Note: These types of questions can be solved using the formula used for finding the interior angle of a regular polygon. Also in the case of polygons, the sum of the measure of exterior angles will be equal to ${360^\circ }$. Always take the angle given equal to $\dfrac{{180^\circ \left( {n - 2} \right)}}{n}$.