Exterior Angles of Polygon

Introduction

Exterior angles of a polygon are formed when by one of its side and extending the other side. Polygon is one of the essential & fundamental shape in geometry. A polygon is a closed two-dimensional geometrical figure with three or more sides. The Greek word polygon is formed from two words ‘poly’ which means many & ‘gon’ means angles. Some real-life examples of a polygon are a hexagon which has a hexagonal shape, a rectangular screen of a laptop, the Bermuda Triangle, Egypt’s pyramid etc. Triangle is widely used in modern construction. Polygon has two types of angles: exterior angles & interior angles. Interior angles are formed inside the polygon, whereas exterior angles are formed outside of a polygon.

Polygons

• A polygon is a closed two-dimensional geometrical figure with two or more sides.

• As mentioned before, the polygon is close-shaped, with no end left open.

• The length of sides & angles of polygons may or may not be the same.

• Triangles, pentagons & hexagons are some examples of polygons.

• Polygons are classified on the basis of the length of sides, angles & no. of sides.

Classification of polygons on the basis of no. of sides −

Sr.No	Name of polygon	No. of sides
1	Triangle	3
2	Quadrilateral	4
3	Pentagon	5
4	Hexagon	6
5	Heptagon	7
6	Octagon	8
7	Nonagon	9
8	Decagon	10
9	Hendecagon	11
10	Dodecagon	12

lassification of polygons on the basis of length of sides & angles −

• On the basis of length of sides polygons are classified into two types

  • Regular polygon

  • Irregular polygon

Regular polygon

• If sides & angles of a polygon are equal then those polygons are said to be a regular polygon.

• These polygons are said to be equiangular & equilateral.

• For example, All sides & angles of a square are equal hence square is a regular polygon.

Irregular polygon

• If the sides & angles of a polygon are unequal then those polygons are said to be an irregular polygon.

• For example, A scalene triangle is the best example of irregular polygon sides & angles of scalane triangle are unequal

Classification of polygons of the basis of angles of polygons

• On the basis of angles polygons are classified into two types convex polygon & concave polygons.

Convex polygon

• If the measure of interior angles is always less than 180⁰, then those polygons are said to be convex polygons.

• Vertices of convex polygons always pointed outward direction.

• For example, hexagon, pentagon & heptagon are some examples of a convex polygon.

Concave polygon

• Concave polygons are exactly the opposite of convex polygons.

• In a concave polygon, the measure of one angle is greater than 180⁰ .

• Vertices of the concave polygon are always pointed in both inward & outward directions.

Regular polygons
Irregular polygons
Convex polygons
Concave polygon

The above figure shows different types of polygons.

Angles of polygons

• As mentioned before, there are two types of angles found in polygons they are given as follows −

  • Interior angles

  • Exterior angles

Interior angles of polygons

Interior angles of any objects which are formed inside the object. Interior angles of the polygon are angles formed inside between two adjacent sides of the polygon. The number of interior angles is equal to the number of sides.

The formula for calculating the interior angle is given as

$$\mathrm{Interior\: angle = 180^0×(n-2)}$$

Where n is the side of a polygon.

Measure of interior angle $\mathrm{=\frac{180^0-n}{n}}$

Exterior angles of the polygon

These angles are formed outside of the polygon. Angles are shown in the above figure. After extending the side of a polygon angle formed by the adjacent side & the extension is known as exterior angles.

Suppose you start moving from the vertex at $\mathrm{\angle 1}$ in a clockwise direction you covered a turn through $\mathrm{\angle 2, \angle 3, \angle 4\: \&\: \angle 5}$ to come back to the same vertex. One complete turn is equal to 360⁰. Hence, the sum of exterior angles of any polygon is 360⁰.

The measure of exterior angle can be calculated by simply dividing no. of sides to 360⁰. The formula is given as,

$$\mathrm{Measure\: of\: exterior\: angle\: = \frac{360^0}{n}}$$

Exterior angles of polygon theorem

Statement: In the case of a convex polygon, the sum of the exterior angles of the polygon, considering each angle is 360⁰.

Proof: Consider a polygon having n number of sides. Also, the sum of exterior angles is N, For any closed shape, the sum of exterior angles is equal to the sum of linear pairs & interior angles.

$$\mathrm{∴ N=180n-180(n-2)}$$

$$\mathrm{N=180n-180n+360}$$

$$\mathrm{N=360}$$

Hence the sum of the exterior angles of a polygon is 360⁰.

Solved examples

1) Calculate the measure of each exterior angle of a regular pentagon.

Ans: Exterior angle $\mathrm{=\frac{360}{n}}$

In the case of pentagon n = 5

Exterior angle = $\mathrm{\frac{360}{5}=72^0}$

Therefore the measure of each exterior angle of the pentagon is 72⁰.

2) Find the measure of each exterior angle of a regular polygon of 18 sides.

Ans: Here n= 18

Exterior angle = $\mathrm{\frac{360}{n}=\frac{360}{18}=20^0}$

Therefore, the measure of each exterior angle of 18 sides polygon is 20⁰.

3) If the sum of the interior angles of the polygon is 2160⁰, find the number of sides of a polygon?

Ans: Sum of the interior angle of polygon = (n-2)×180⁰

$$\mathrm{2160^0 = (n-2)×180^0}$$

$$\mathrm{\frac{2160^0}{180^0}=(n-2)}$$

$$\mathrm{(n-2) = 12}$$

$$\mathrm{n = 14}$$

Therefore the number of sides of the polygon is 14.

Conclusion

In this tutorial we have discussed polygon, angles of a polygon & exterior angles of polygons with solved examples. In geometry, the polygon is a two-dimensional closed shape with three or more than three sides. Polygons are classified on the basis of length of sides, angles & no. of sides. In a polygon, there are two types of angles that can be found that are interior angles & exterior angles. Also, we have studied formulas for calculating the interior & exterior angles of a polygon.

FAQs

1. What are remote interior angles?

An angle that is not adjacent to the exterior angle & doesn’t share a vertex of the exterior angle is known as the remote interior angle.

2. What is the measure of the angles of the hexagon?

The measure of each interior & exterior angle of the hexagon are 120⁰ & 60⁰respectively.

3. Can a diamond be a polygon?

Yes. diamond is a quadrilateral. Quadrilateral is a type of polygon, therefore, all diamonds are polygon.

4. Do all polygons have exterior angles that add upto 360⁰?

The sum of exterior angles of any type of polygon is 360⁰.

5. What is exterior angle property?

In a triangle, the measure of an exterior angle is equal to the sum of the measure of two interior angles.