Class-8 Polygon

Introduction to Polygon

Types of Polygons

Concave Polygon

Convex Polygon

Regular Polygon

Irregular Polygon

Sum of Angles of a Polygon

Sum of Exterior Angles of a Polygon

Number of Diagonals of a Polygon

Polygon MCQ

Polygon Worksheets

Answer Sheet

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 = 10²⁄₇

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°