# Exterior angle of Polygon

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.

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.