The exterior angles of polygons are formed when we extend the sides of a polygon. The sum total of these angles is always equal to 360°. Therefore, if the polygon is regular, we can divide 360° for the number of sides to find the measure of an exterior angle of the polygon. If the polygon is irregular, we need to use other methods to find the measures of each angle.

Here, we will learn more about the exterior angles of polygons.

## Sum of exterior angles of a polygon

The sum of the exterior angles of any polygon is always equal to 360°. This property applies regardless of whether the polygon is regular or irregular. For example, in the following diagram, we can look at the exterior angles of a pentagon.

We can see that when we put them together, the five exterior angles form a circle. This represents one complete turn, that is, an angle of 360°.

Now, let’s look at the following hexagon with its exterior angles.

In the same way, we can observe that when we join them, we form a complete angle of 360°.

The sum is always equal to 360°. This means that as the number of sides of the polygon increase, the measures of the individual exterior angles get smaller.

## Exterior angles of a regular polygon

A regular polygon is a geometric figure that has all its sides with the same length and all its interior angles with the same measure. This means that all of its exterior angles also have the same measure.

Since the sum of exterior angles of any polygon is always equal to 360°, we can divide by the number of sides of the regular polygon to get the measure of the individual angles.

For example, for a pentagon, we have to divide 360° by 5:

360°÷5 = 72°

Each exterior angle in a regular pentagon measures 72°. In the following table, we can see the exterior angle measures of some common regular polygons.

Polygon | Each angle |

Triangle | 120° |

Square | 90° |

Pentagon | 72° |

Hexagon | 60° |

Heptagon | 51.43° |

Octagon | 45° |

## How to calculate exterior angle measures of irregular polygons?

We can determine the measure of a missing exterior angle if we know the measures of the other exterior angles. For this, we have to add all the known angles and subtract from 360°.

For example, if we have the exterior angles 60°, 70°, 80°, and 85° in a pentagon, we have to start by determining their sum and then subtract it from 360°:

60°+70°+80°+85°=295°

⇒ 360°-295°=65°

The measure of the missing angle is 65°.

Additionally, we can also calculate the exterior angle measures if we know the interior angle measures. For this, we consider that the sum of an interior angle and its corresponding exterior angle is equal to 180°.

Therefore, we can subtract the interior angle from 180° to find the measure of the exterior angle.

For example, if we have interior angles 90°, 120°, 110°, 105°, and 115° in a pentagon, we have to subtract each 180° angle to find the corresponding exterior angles:

180°-90°=90°

180°-120°=60°

180°-110°=70°

180°-105°=75°

180°-115°=65°

Therefore, the measures of the exterior angles of the pentagon are 90°, 60°, 70°, 75°, and 65°.

## Solved examples of exterior angles of polygons

### EXAMPLE 1

Determine the measures of the missing exterior angles in the pentagon below.

**Solution:** The angles that have the same color and are represented with a double line are equal. Therefore, we have *a*=60°. To find the measure of angle *b*, we have to add the known angle measures and subtract from 360°. Therefore, we have:

80°+60°+60°+90° = 290°

⇒ 360°-290° = 70°

Angle *b* measures 70°.

### EXAMPLE 2

Find the missing exterior angle measures in the hexagon below.

**Solution:** The angles marked with a double line and that have the same color share the same measure, so we have *a*=50°. To find the measure of angle *b*, we have to add the known angle measures and subtract the result from 360°. Therefore, we have:

50°+50°+70°+75°+60° = 305°

⇒ 360°-305° = 55°

The measure of angle *b* is 55°.

### EXAMPLE 3

Find the measures of the exterior angles of the pentagon.

**Solution:** We have to subtract each corresponding interior angle from 180° to find the exterior angle measures. Therefore, we have:

180°-110° = 70°

180°-120° = 60°

180°-100° = 80°

180°-90° = 90°

Now, we have a missing angle. We can find that angle by adding the known angles and subtracting from 360°:

70°+60°+80°+90° = 300°

360°-300° = 60°

### EXAMPLE 4

Determine the measures of the exterior angles in the hexagon below.

**Solution:** In this case, we have the measures of the interior angles. Therefore, we subtract each 180° angle to find the measure of the corresponding exterior angle:

180°-110°=70°

180°-120°=60°

180°-130°=50°

180°-125°=55°

180°-100°=80°

To find the measure of the missing exterior angle, we can add the measures of the known exterior angles and subtract from 360°. Therefore, we have:

70°+60°+50°+55°+80°=315°

⇒ 360°-315°=45°

## See also

Interested in learning more about angles of polygons? Take a look at these pages:

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