In this chapter we will learn about the concept of exterior angle of polygon with some examples.

## What are exterior angles of Polygon ?

The angle which** lies between polygon’s arm on one side and extended arm on other side** is called **Exterior angle**.

Exterior angles are formed by side and adjacent extended side of the given polygon.

**For example**, observe the below triangle ABC.

Here ∠1 is the exterior angle formed between side BC and extended side AB.

Other points related to exterior angles are:

(a) Exterior angle is formed outside the shape of given polygon.

(b) There are as many exterior angles as there are interior angles in polygon.

### Exterior angle formula for regular polygons

You can calculate the measure of exterior angle by using the following formula;

\mathtt{Exterior\ angle\ =\ \frac{360\ }{n}}

Where;

n = number of side of polygon

External angle **can also be calculated with the help of internal angle**.

In any polygon, the sum of interior angle & exterior angle **always form a straight line**.

So we can say that;**External Angle + Internal angle = 180 degree**

**For Example;**

Consider the below triangle ABC;

Here,

∠1 ⟹ exterior angle

∠2 ⟹ interior angle

See that both ∠1 and ∠2 form straight line.

So;

∠1 + ∠2 = 180 degree

∠1 + 60 = 180

∠1 = 120 degree

Hence, the above external angle forms 120 degree angle.

This technique is also helpful to find value of exterior angle of any polygon.

### Sum of all exterior angle of polygon

The **sum of all exterior angle of polygon always measure 360** **degree**.

Let us prove the above statement.

Take a regular triangle ABC.

**In the above figure;**

∠1, ∠2 and ∠3 are the interior angles.

∠4, ∠5 and ∠6 are the exterior angles.

**Note that;**

∠1 + ∠5 = 180 degree { straight line angle}

**Similarly;**

∠2 + ∠6 = 180

∠3 + ∠4 = 180

**Adding all the three equation, we get;**

∠1 + ∠5 + ∠2 + ∠6 + ∠3 + ∠4 = 180+180+180

∠1 + ∠5 + ∠2 + ∠6 + ∠3 + ∠4 = 540 -eq (1)

**We know that sum of angle of triangle measure 180 degree.**

∠1 + ∠2 + ∠3 = 180

Putting above expression in eq (1)

180 + ∠4 + ∠5 + ∠6 = 540

∠4 + ∠5 + ∠6 = 540 – 180

∠4 + ∠5 + ∠6 = 360

Hence, **the sum of exterior angle of triangle is 360.**

## Calculating external angles of polygon

I hope you understood the above concepts. In this section we will individually calculate external angle values of different regular polygons.

### Exterior angles of Triangle

Triangle is a polygon with three sides and three external angles.

Given below is the image of regular triangle where;

∠1, ∠2 and ∠3 are the interior angles.

∠4, ∠5 and ∠6 are the exterior angles.

We know that;

\mathtt{Exterior\ angle\ =\ \frac{360\ }{n}}\\\ \\ \mathtt{Exterior\ angle\ =\ \frac{360}{3}}\\\ \\ \mathtt{Exterior\ angle\ =\ 120}

Hence, in triangle exterior angle measures **120 degree**.

### Exterior angle of quadrilateral

It’s a polygon with 4 internal angle and 4 external angle.

∠1, ∠2, ∠3 and ∠4 are the interior angles.

∠5, ∠6, ∠7, ∠6 are the exterior angles.

We know that;

\mathtt{Exterior\ angle\ =\ \frac{360\ }{n}}\\\ \\ \mathtt{Exterior\ angle\ =\ \frac{360}{4}}\\\ \\ \mathtt{Exterior\ angle\ =\ 90}

Hence, in quadrilateral exterior angle **measures 90 degree.**

### Exterior angle of Pentagon

It’s a polygon with 5 external and internal angles.

Given below is the regular pentagon ABCDE with external angles ∠1, ∠2, ∠3, ∠4 and ∠5.

We know that;

\mathtt{Exterior\ angle\ =\ \frac{360\ }{n}}\\\ \\ \mathtt{Exterior\ angle\ =\ \frac{360}{5}}\\\ \\ \mathtt{Exterior\ angle\ =\ 72\ degree}

Hence in a regular pentagon, the exterior angle measures **72 degree.**

### Exterior angle of Hexagon

It’s a polygon with 6 sides.

Given below is the regular hexagon ABCDEF with angles ∠1, ∠2, ∠3, ∠4, ∠5 and ∠6.

We know that;

\mathtt{Exterior\ angle\ =\ \frac{360\ }{n}}\\\ \\ \mathtt{Exterior\ angle\ =\ \frac{360}{6}}\\\ \\ \mathtt{Exterior\ angle\ =\ 60\ degree}

Hence, in a regular hexagon,** each exterior angle measures 60 degree**.

### Exterior angle of Heptagon

Heptagon is a polygon with 7 sides.

Given below is the heptagon ABCDEFG with exterior angles ∠1, ∠2, ∠3, ∠4, ∠5 , ∠6 and ∠7.

We know that;

\mathtt{Exterior\ angle\ =\ \frac{360\ }{n}}\\\ \\ \mathtt{Exterior\ angle\ =\ \frac{360}{7}}\\\ \\ \mathtt{Exterior\ angle\ =\ 51.42\ degree}

Hence, exterior angle of **regular heptagon measures 51.42 degree**.

### Exterior angle of Octagon

Octagon is a polygon with 8 sides.

Given below is a regular octagon with exterior angles ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7 and ∠8.

We know that;

\mathtt{Exterior\ angle\ =\ \frac{360\ }{8}}\\\ \\ \mathtt{Exterior\ angle\ =\ \frac{360}{8}}\\\ \\ \mathtt{Exterior\ angle\ =\ 45\ degree}

Hence in a regular octagon each exterior angle measure **45 degree.**