# 5.28: Exterior Angles in Convex Polygons

- Page ID
- 5013

Measure of angles on the outside of a polygon formed by extending a side.

## Exterior Angle Sum Theorem

An **exterior angle** is an angle that is formed by extending a side of the polygon.

As you can see, there are two sets of exterior angles for any vertex on a polygon, one going around clockwise (1st hexagon), and the other going around counter-clockwise (2nd hexagon). The angles with the same colors are vertical and congruent.

The **Exterior Angle Sum Theorem** states that the sum of the exterior angles of ANY convex polygon is \(360^{\circ}\). If the polygon is regular with n sides, this means that each exterior angle is \(\dfrac{360^{\circ}}{n}\).

What if you were given a seven-sided **regular polygon**? How could you determine the measure of each of its exterior angles?

Example \(\PageIndex{1}\)

What is the measure of each exterior angle of a regular 12-gon?

**Solution**

Divide \(360^{\circ}\) by the given number of sides.

\(30^{\circ}\)

Example \(\PageIndex{2}\)

What is the measure of each exterior angle of a regular 100-gon?

**Solution**

Divide \(360^{\circ}\) by the given number of sides.

\(3.6^{\circ}\)

Example \(\PageIndex{3}\)

What is y?

**Solution**

\(y\) is an exterior angle and all the given angles add up to \(360^{\circ}\). Set up an equation.

\(\begin{aligned} 70^{\circ}+60^{\circ}+65^{\circ}+40^{\circ}+y&=360^{\circ} \\ y&=125^{\circ} \end{aligned}\)

Example \(\PageIndex{4}\)

What is the measure of each exterior angle of a regular heptagon?

**Solution**

Because the polygon is regular, the interior angles are equal. It also means the exterior angles are equal. \(\dfrac{360^{\circ}}{7}\approx 51.43^{\circ}\)

Example \(\PageIndex{5}\)

What is the sum of the exterior angles in a regular 15-gon?

**Solution**

The sum of the exterior angles in any convex polygon, including a regular 15-gon, is \(360^{\circ}\).

## Review

- What is the measure of each exterior angle of a regular decagon?
- What is the measure of each exterior angle of a regular 30-gon?
- What is the sum of the exterior angles of a regular 27-gon?

Find the measure of the missing variables:

- The exterior angles of a quadrilateral are \(x^{\circ}\), \(2x^{\circ}\), \(3x^{\circ}\), and \(4x^{\circ}\). What is \(x\)?

Find the measure of each exterior angle for each regular polygon below:

- octagon
- nonagon
- triangle
- pentagon

## Review (Answers)

To see the Review answers, open this PDF file and look for section 6.2.

## Resources

## Vocabulary

Term | Definition |
---|---|

exterior angle |
An angle that is formed by extending a side of the polygon. |

regular polygon |
A polygon in which all of its sides and all of its angles are congruent. |

Exterior Angle Sum Theorem |
Exterior Angle Sum Theorem states that the exterior angles of any polygon will always add up to 360 degrees. |

## Additional Resources

Interactive Element

Video: Interior and Exterior Angles of a Polygon

Activities: Exterior Angles in Convex Polygons Discussion Questions

Study Aids: Polygons Study Guide

Real World: Exterior Angles Theorem