# Exterior Angles of a Polygon

Last Updated : 16 May, 2024

Polygon is a closed, connected shape made of straight lines. It may be a flat or a plane figure spanned across two-dimensions. A polygon is an enclosed figure that can have more than 3 sides. The lines forming the polygon are known as the edges or sides and the points where they meet are known as vertices. The sides that share a common vertex among them are known as adjacent sides. The angle enclosed within the adjacent side is called the interior angle and the outer angle is called the exterior angle.

## What is Exterior Angle?

An exterior angle basically is formed by the intersection of any of the sides of a polygon and extension of the adjacent side of the chosen side. Interior and exterior angles formed within a pair of adjacent sides form a complete 180 degrees angle.Â

## Measures of Exterior Angles

We can measure exterior angles using following steps:

1. They are formed on the outer part, that is, the exterior of the angle.
2. The corresponding sum of the exterior and interior angle formed on the same side = 180Â°.
3. The sum of all the exterior angles of the polygon is independent of the number of sides and is equal to 360Â°, because it takes one complete turn to cover polygon in either clockwise or anti-clockwise direction.

### Exterior Angle of Regular Polygon Formula

For a regular polygon (a polygon with all sides and angles equal), the exterior angle can be calculated using the formula:

Exterior Angle = 360Â°/n

Where n is the number of sides of polygon.

## Theorem for Exterior Angles Sum of a Polygon

The sum of the exterior angles of any polygon, one at each vertex, is always 360Â°.

### Proof

Let us consider a polygon which has n number of sides. The sum of the exterior angles is N.

The sum of exterior angles of a polygon(N) =Â Difference between {the sum of the linear pairs (180n)} – {the sum of the interior angles.(180(n – 2))}

â‡’ N = 180n âˆ’ 180(n – 2)
â‡’ N = 180n âˆ’ 180n + 360
â‡’ N = 360

Hence, we have the sum of the exterior angle of a polygon is 360Â°.

## Exterior and Interior Angles of Polygon

The key differences between interior and exterior angles in any polygon are listed in the following table:

Aspect Interior Angles Exterior Angles
Definition Angles inside the polygon, formed by two adjacent sides. Angles formed between one side of the polygon and the extension of an adjacent side.
Sum in a Polygon (nâˆ’2) Ã— 180Â° for an n-sided polygon. Always 360Â° for any polygon.
Individual Angle in a Regular Polygon InteriorÂ Angle =(n âˆ’ 2) Ã— 180Â°â€‹
Where n is the number of sides.
ExteriorÂ Angle = 360Â°/n
Where n is the number of sides.
Relationship to Each Other Supplementary to the exterior angle. Supplementary to the interior angle.
Visual Representation Found inside the polygon. Found outside the polygon, adjacent to each interior angle.

## Sample Problems on Exterior Angles

Example 1: Find the exterior angle marked with x.Â

Solution:

Since the sum of exterior angles is 360 degrees, the following properties hold:

âˆ 1 + âˆ 2 + âˆ 3 + âˆ 4 + âˆ 5 = 360Â°
â‡’ 50Â° + 75Â° + 40Â° + 125Â° + x = 360Â°
â‡’ x = 360Â°

Example 2: Determine each exterior angle of the quadrilateral.

Solution:

Since, it is a regular polygon, where all interior and exterior angles are equal.

Thus, measure of each exterior angle = 360Â°/ Number of sides
â‡’ Measure of each exterior angle = 360Â°/4
â‡’ Measure of each exterior angle = 90Â°

Example 3: Find the regular polygon where each of the exterior angle is equivalent to 60Â°.Â

Solution:

Since it is a regular polygon, the number of sides can be calculated by the sum of all exterior angles, which is 360 degrees divided by the measure of each exterior angle.Â

Number of sides = (Sum of all exterior angles of a polygon)/n
â‡’ Value of one pair of side = 360Â°/60Â° = 6

Therefore, this is a polygon enclosed within 6 sides, that is hexagon.

Example 4: Find the interior angles ‘x, y’, and exterior angles ‘w, z’ of this polygon?

Solution:

Here we have âˆ DAC = 110Â° that is an exterior angle and âˆ ACB = 50Â° that is an interior angle.

Firstly we have to find interior angles ‘x’ and ‘y’.
âˆ DAC + âˆ x = 180Â° Â {Linear pairs}
â‡’ 110Â° Â + âˆ x = 180Â° Â
â‡’ âˆ x = 180Â° – 110Â° Â
â‡’ âˆ x = 70Â°

Now,Â
âˆ x + âˆ y + âˆ ACB = 180Â° {Angle sum property of a triangle}Â
â‡’ 70Â°+ âˆ y + 50Â° = 180Â° Â
â‡’ âˆ y + 120Â° = 180Â°Â
â‡’ âˆ y = 180Â° – 120Â°Â
â‡’ âˆ y = 60Â°Â

Secondly now we can find exterior angles ‘w’ and ‘z’.
âˆ w + âˆ ACB = 180Â° {Linear pairs}
â‡’ âˆ w + 50Â° = 180Â°Â
â‡’ âˆ w = 180Â° – 50Â°Â
â‡’ âˆ w = 130Â°Â

Now we can use the theorem exterior angles sum of a polygon,
âˆ w + âˆ z Â + âˆ DAC = 360Â° {Sum of exterior angle of a polygon is 360Â°}
â‡’ 130Â° + âˆ z + 110Â° = 360Â°
â‡’ 240Â° + âˆ z = 360Â°
â‡’ âˆ z = 360Â° – 240Â°
â‡’ âˆ z = 120Â°

## FAQs on Exterior Angles of a Polygon

### Define exterior angles of a polygon.

Exterior angles of a polygon are the angles formed between one side of the polygon and the extension of an adjacent side.

### How do you calculate the exterior angle of a regular polygon?

We can calculate the exterior angles of any regular polygon using:

Exterior Angle = 360Â°/n

Where n is the number of sides of polygon.

### What is the sum of the exterior angles of any polygon?

The sum of the exterior angles of any polygon, regardless of the number of sides, is always 360Â°.

### Why is the sum of the exterior angles always 360Â°?

The sum of the exterior angles of a polygon is always 360Â° because they effectively represent one complete turn around the polygon.

### How do exterior angles relate to interior angles?

Each exterior angle of a polygon is supplementary to its corresponding interior angle. This means that the exterior angle and interior angle add up to 180Â°.

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