Exterior Angles in Convex Polygons

Angles on the outside of a polygon formed by extending a side.

%
Progress

MEMORY METER
This indicates how strong in your memory this concept is
Progress
%
Exterior Angles in Convex Polygons

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

Exterior Angles in Convex Polygons

Recall that an exterior angle is an angle on the outside of a polygon and 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. It does not matter which set you use because one set is just the vertical angles of the other, making the measurement equal. In the picture above, the color-matched angles are vertical angles and congruent. The Exterior Angle Sum Theorem stated that the exterior angles of a triangle add up to . Let’s extend this theorem to all polygons.

Watch the second half of this video.

Investigation: Exterior Angle Tear-Up

Tools Needed: pencil, paper, colored pencils, scissors

1. Draw a hexagon like the hexagons above. Color in the exterior angles as well.
2. Cut out each exterior angle and label them 1-6.
3. Fit the six angles together by putting their vertices together. What happens?

The angles all fit around a point, meaning that the exterior angles of a hexagon add up to , just like a triangle. We can say this is true for all polygons.

Exterior Angle Sum Theorem: The sum of the exterior angles of any polygon is .

Proof of the Exterior Angle Sum Theorem:

Given: Any gon with sides, interior angles and exterior angles.

Prove: exterior angles add up to

NOTE: The interior angles are .

The exterior angles are .

Statement Reason
1. Any gon with sides, interior angles and exterior angles. Given
2. and are a linear pair Definition of a linear pair
3. and are supplementary Linear Pair Postulate
4. Definition of supplementary angles
5. Sum of all interior and exterior angles in an gon
6. Polygon Sum Formula
7. Substitution PoE
8. Distributive PoE
9. Subtraction PoE

Solving for Unknown Angle Measurments

What is ?

is an exterior angle, as well as all the other given angle measures. Exterior angles add up to , so set up an equation.

Measuring Exterior Angles

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

Because the polygon is regular, each interior angle is equal. This also means that all the exterior angles are equal. The exterior angles add up to , so each angle is .

Calculating the Sum of Exterior Angles

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

The sum of the exterior angles in any convex polygon, including a regular 15-gon, is .

Earlier Problem Revisited

The exterior angles of a regular polygon sum to . The measure of each exterior angle in a dodecagon (twelve-sided regular polygon) is .

Examples

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

Example 1

12-gon

For each, divide by  and by the given number of sides.

100-gon

36-gon

Review

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

Find the measure of the missing variables:

1. The exterior angles of a quadrilateral are and What is ?

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

1. octagon
2. nonagon
3. triangle
4. pentagon
5. 50-gon
6. heptagon
7. 34-gon
8. Challenge Each interior angle forms a linear pair with an exterior angle. In a regular polygon you can use two different formulas to find the measure of each exterior angle. One way is and the other is ( minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent.
9. Angle Puzzle Find the measures of the lettered angles below given that .

Review (Answers)

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

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Vocabulary Language: English

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

Explore More

Sign in to explore more, including practice questions and solutions for Exterior Angles in Convex Polygons.
Please wait...
Please wait...