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5.28: Exterior Angles in Convex Polygons

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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.

f-d_93c55f216c91786218045f3400bcd8e12ff0fc8874fea71333700992+IMAGE_TINY+IMAGE_TINY.png
Figure 5.28.1

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. If the polygon is regular with n sides, this means that each exterior angle is 360n.

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

Example 5.28.1

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

Solution

Divide 360 by the given number of sides.

30

Example 5.28.2

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

Solution

Divide 360 by the given number of sides.

3.6

Example 5.28.3

What is y?

f-d_aa7748309ebb648d121733e41ca0653dc76b0f6fbf857960e96912ae+IMAGE_TINY+IMAGE_TINY.png
Figure 5.28.2

Solution

y is an exterior angle and all the given angles add up to 360. Set up an equation.

70+60+65+40+y=360y=125

Example 5.28.4

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. 360751.43

Example 5.28.5

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.

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. f-d_a1152bf181eba513d57b67d135f8e7c0df339fe680e57f42eb7d0f7d+IMAGE_TINY+IMAGE_TINY.png
    Figure 5.28.3
  2. f-d_69769e9c68321bcecf0ab8080080b54410313cc40badd27aeee383ba+IMAGE_TINY+IMAGE_TINY.png
    Figure 5.28.4
  1. The exterior angles of a quadrilateral are x, 2x, 3x, and 4x. What is x?

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

  1. octagon
  2. nonagon
  3. triangle
  4. 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


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