# Class-8 Polygon

Sum of Exterior Angles of a Polygon

Number of Diagonals of a Polygon

## Introduction to Polygon

A simple closed curve made up of only line segments is known as a polygon. The line segments are called sides of a polygon. The line segments forming a polygon intersect only at end points and each end point is shared by only two-line segments. Figure below shows various types of polygons.

Triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon and decagon can be called as polygon.

## Types of Polygons

There are two types of polygons.

- Concave Polygon
- Convex Polygon

## Concave Polygon

A polygon in which at least one angle is greater than 180° is known as a concave polygon. Below given figure is an example of concave polygon, here ∠ABC is more than 180°.

## Convex Polygon

A polygon in which each angle is less than 180° is known as a convex polygon. In the below figure ABCD is a convex polygon.

## Regular Polygon

A polygon having all sides equal and all angles equal is known as a regular polygon. Equilateral triangle and square are examples of regular polygons.

## Irregular Polygon

Polygons which do not follow regular polygons properties are known as irregular polygon. Rectangle and rhombus are examples of irregular polygons.

## Sum of Angles of a Polygon

If we draw all possible diagonals through a single vertex of a polygon to form as many triangles as possible, then the number of triangles can be formed is two less than the number of sides in the polygon.

So, if a polygon has 'n' sides, then the number of triangle formed will be n − 2.

As we know, the sum of angles of a triangle = 180°

Sum of angles of a polygon having 'n' sides = (n − 2) × 180°

Each interior angle = ^{{(n − 2) × 180°}}⁄_{n}

**Example 1.** Find the measure of each exterior angle of a regular polygon having 9 sides.

**Solution.** Each exterior angle of a regular polygon having 9 sides = ^{360°}⁄_{9} = 40°

**Example 2.** Find the measure of each interior angle of a regular hexagon.**Solution.** Each exterior angle of a regular hexagon = ^{360°}⁄_{6} = 60°

Each interior angles of the hexagon = 180° − 60° = 120°

**Example 3.** Is it possible to have a regular polygon having each exterior angle 35o.

**Solution.** Let's assume the regular polygon is having 'n' sides.

Each exterior angle = ^{360°}⁄_{n}

=> 350 = ^{360°}⁄_{n}

=> n = ^{360°}⁄_{35°}

=> n = 10^{2}⁄_{7}

**Example 4.** If each interior angle of a regular ploygon is 156°, then find the number of sides in it.

**Solution.** Let's assume the number of sides of the regular ploygon is 'n'.

Interior angle = ^{{(n − 2) × 180°}}⁄_{n}

=> 156° = ^{(180n − 360)}⁄_{n}

=> 156n = 180n − 360

=> 24n = 360

=> n = ^{360}⁄_{24}

=> n = 15

Hence, the regular polygon has 15 sides.

**Example 5.** An exterior angle and interior angle of a regular polygon are in the ratio 5 : 4. Find the number of sides in the polygon.

**Solution.** Exterior angle : Interior angle = 5 : 4

In other words, we can say exterior angle is 5x and interior angle is 4x.

As we know, exterior angle + interior angle = 180°

5x + 4x = 180

=> 9x = 180

=> x = 20

Exterior angle = 5x = 100°

Interior angle = 4x = 80°

## Sum of Exterior Angles of a Polygon

If the sides of a polygon are extended in order, the sum of exterior angles formed is always equal to 360°.

∠1 + ∠2 + ∠3 + ∠4 + ∠5 = 360°

## Number of Diagonals of a Polygon

Number of diagonals in a polygon having 'n' sides = ^{n(n − 3)}⁄_{2}

## Class-8 Polygon Test

## Class-8 Polygon Worksheets

## Answer Sheet

**Polygon-Answer**Download the pdf

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