Most of the students will feel theorems are very difficult to understand and remember the concept. But we will help you to learn about the Rectilinear Figures here. In this article, we will discuss the Sum of the Exterior Angles of an n-sided Polygon Theorem with step by step explanation. By learning the theorems you can solve different types of problems in exams.

- The measure of each exterior angle is 360°/n
- Sum of exterior angles of a polygon is ∠a’ + ∠b’ + ∠c’ + ….. + ∠n’ = 360°.
- The sum of the exterior angles is 360°.

## Sum of the Exterior Angles of an n-sided Polygon

**Theorem Statement:**

If the sum of a convex polygon is produced in some order the sum of all the exterior angles is equal to four right angles.

Given

Let PQRS……. Z be a convex polygon of n sides have been produced in the same order

To prove that:

The sum of the exterior angles is 4 right angles they are ∠p’ + ∠q’ + ∠r’ + ……..+ ∠z’ = 4 × 90° = 360°

Proof:

∠p + ∠p’ = 2 right angles

Similarly

∠q + ∠q’ = 2 right angles

∠z + ∠z’ = 2 right angles [ they are in the linear form]

(∠p + ∠q + ∠r +…….∠n) + (∠p’ + ∠q’ + ∠r’ +……∠z’) = 2n right angles [ the polygon has n sides]

(2n – 4) right angles + (∠p + 3∠p + ∠q + ∠r +…..∠q’ + ∠r’ + …..∠z’) = 2n right angles + ∠z’ = (2z – 4) right angles

∠p’ + ∠q’ + ∠r’ +……+∠z’ [ from above equation]

[2n – (2n – 4)] right angles

4 right angles

4 × 90°

360°

Hence proved

**Points to remember:**

1. In an N-sided regular polygon each exterior angle is 360 degrees/n.

2. The greater the number of sides of a regular polygon, the greater is the value of each interior angle and the smaller is the value of each exterior angle.

3. If each exterior angle of a regular polygon is x°, the polygon has 360/x sides.

Also, Check: Sum of the Interior Angles of an n-sided Polygon

### Problems on Sum of the Exterior Angles of an n-sided Polygon

**Example 1.**

Find the measures of each exterior angle of a regular hexagon

**Solution:**

Given that

Regular hexagon = 6 sides

Here n = 6

Each exterior angles = 360/n

= 360/6

= 60

Therefore the measure of each exterior angle of a regular hexagon is 60 degrees.

**Example 2.**

Find the number of sides of a regular polygon if each of its exterior angles is 20°

**Solution:**

Here exterior angles x = 20°

Number of sides = 360/20 = 18

Therefore there are 18 sides of the regular polygon

**Example 3.**

Find the number of sides of a regular polygon if each of its interior angles is 120°

**Solution:**

Each interior angle = 120°

Therefore each exterior angle = 180° – 120° = 40°

We know that the total number of Sides of a regular polygon is 360/x where each exterior angle is x°

Number of sides = 360°/40 = 90°

Therefore there are 9 sides of a regular polygon.

**Example 4.**

Find the measures of each exterior angles of a regular square

**Solution:**

Given that

Regular square = 4 sides

Here n = 4

Each exterior angles = 360/n

= 360/4

= 90

Therefore the measure of each exterior angle of a regular hexagon is 90 degrees.

**Example 5.**

Find the number of sides of a regular polygon if each of its exterior angles is 60°

**Solution:**

Here exterior angles x = 60°

Number of sides = 360/60 = 6

Therefore there are 6 sides of the regular polygon

### FAQs on Sum of the Exterior Angles of an n-sided Polygon

**1. How do you find the sum of the exterior angles of a polygon?**

The sum of the exterior angles of a regular polygon will always equal 360 degrees. To find the value of a given exterior angle of a regular polygon, simply divide 360 by the number of sides or angles that the polygon has.

**2. What is the angle of N sided polygon?**

In a polygon of ‘n’ sides, the sum of the interior angles is equal to (2n – 4) × 90°

**3. What is the sum of N-sided polygon?**

In a regular polygon of n sides, all angles are equal. Therefore, each interior angle = (2n−4)×90°n.