# Exterior Angles of a Polygon: Sum & Formula

TL;DR  
The exterior angles of a polygon are the angles between each side and the extension of its neighbour, and they always sum to 360°. In a regular polygon, each exterior angle is \( \frac{360°}{n} \). This article defines the exterior angle, derives the 360° sum, gives the formula, and works through examples.

## What Is an Exterior Angle of a Polygon?

An **exterior angle** of a polygon is the angle formed _outside_ the polygon between one side and the **extension** of the side next to it. Each vertex has an exterior angle that pairs with the **interior angle** at the same vertex.  
Because the side and its extension form a straight line, together they make a straight angle:
\[ \text{interior angle} + \text{exterior angle} = 180°. \]  
So the exterior angle is the **supplement** of the interior angle at the same vertex.

## Why the Exterior Angles Always Sum to 360°

This is the result everything else depends on. Take any polygon with \( n \) sides. Each vertex contributes one interior and one exterior angle, and the two add to \( n \times 180°. \) Now subtract the sum of the interior angles for an \( n \)-sided polygon:
\[ \text{sum of exterior angles} = n \times 180° - (n - 2) \times 180° = 360° .\]  
The 180°n terms cancel, and 360° is all that remains. This holds true for **regular and irregular polygons alike**.

## The Exterior Angle of a Regular Polygon

In a **regular polygon**, each exterior angle is \( \frac{360°}{n} \).  
For example:  
- A regular hexagon has \( n=6 \) and each exterior angle is \( \frac{360°}{6} = 60° \).  
- A square has \( n=4 \) and each exterior angle is \( 90° \).

| Regular polygon       | Sides (n) | Each exterior angle (\( \frac{360°}{n} \))  |
|-----------------------|-----------|-----------------------------------|
| Equilateral triangle   | 3        | 120°                              |
| Square                 | 4        | 90°                               |
| Regular pentagon      | 5        | 72°                               |
| Regular hexagon       | 6        | 60°                               |
| Regular octagon       | 8        | 45°                               |
| Regular decagon       | 10       | 36°                               |

## Examples of Exterior Angles of a Polygon

### Example 1
**Find each exterior angle of a regular octagon.**  
Each exterior angle = \( \frac{360°}{8} = 45° \).

### Example 2
**Find each interior angle of a regular pentagon using its exterior angle.**  
Each exterior angle = \( \frac{360°}{5} = 72° \). The interior angle is its supplement: \( 180° - 72° = 108° \).

### Example 3
**Three exterior angles of a quadrilateral are 80°, 95°, and 70°. Find the fourth.**  
The fourth angle is \( 360° - (80° + 95° + 70°) = 115° \).

### Example 4
**A regular polygon has each exterior angle equal to 40°. How many sides does it have?**  
Use \( n = \frac{360°}{40°} = 9 \).

### Example 5
**A regular polygon has each interior angle 150°.**  
Each exterior angle is \( 180° - 150° = 30° \). Then \( n = \frac{360°}{30°} = 12 \).

### Example 6
**Can a regular polygon have an exterior angle of 50°?**  
Using \( n = \frac{360°}{50°} = 7.2 \), it must be a whole number, so **no**.

## Why Exterior Angles Matter Beyond the Classroom

The constant 360° turn is an important rule that anything following a closed path obeys, with applications in:  
- **Robotics**  
- **Surveying**  
- **Road design**  
- **Tiling**

## Key Takeaways
- The **exterior angles of a polygon** always sum to 360°.  
- For a regular polygon, each exterior angle is \( \frac{360°}{n} \).
- The most common mistake is mixing totals between interior and exterior angles.

## Practice Problems
1. Find each exterior angle of a regular decagon.
2. A regular polygon has each exterior angle 24°. How many sides does it have?
3. Four exterior angles of a pentagon are 60°, 75°, 80°, and 65°. Find the fifth.