# Ex.3.2 Q4 Understanding Quadrilaterals Solution - NCERT Maths Class 8

## Question

How many sides does a regular polygon have if each of its interior angles is \(165^\circ\)?

## Text Solution

**What is Known?**

Measure of an interior angle is \(165^\circ\).

**What is Unknown?**

The number of sides of the regular polygon.

**Reasoning:**

We know that:

a) Irrespective of the number of sides of the polygon, the measure of the exterior angles is equal and the sum of the measures of all the exterior angles of the regular polygon is equal to \(360°\).

b) The measure of interior angle of regular polygon is \(\rm{}\,(n-2) *180/n\), where ‘\(n\)’ is the number of sides of the polygon

**Steps:**

Let number of sides be \(\rm n\).

Measure of each interior angle \(= {\rm{16}}{{\rm{5}}^{\rm{\circ}}}\)

Measure of each exterior angle \(={\rm{18}}0^\circ {\rm{ }} - {\rm{ 165}}^\circ {\rm{ }} = {\rm{ 15}}^\circ \) [linear pair angles]

Number of sides

\[\begin{align}&= \,\,\frac{{{\text{Sum of exterior angles}}}}{{{\text{Each exterior angle}}}}\\&={} \,\,\frac{{{\rm{36}}0^\circ }}{{{\rm{15}}^\circ }}\\&= \,\,\,24\end{align}\]

Hence, the regular polygon has \(24 \) sides.