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Exterior Angles in Convex Polygons

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

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Exterior Angles in Convex Polygons

What if you were given a seven-sided regular polygon? How could you determine the measure of each of its exterior angles? After completing this Concept, you'll be able to use the Exterior Angle Sum Theorem to solve problems like this one.

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CK-12 Exterior Angles in Convex Polygons

Watch the second half of this video.

James Sousa: Angles of Convex Polygons

Guidance

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 (\begin{align*}1^{st}\end{align*} hexagon), and the other going around counter-clockwise (\begin{align*}2^{nd}\end{align*} 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 \begin{align*}360^\circ\end{align*}. If the polygon is regular with \begin{align*}n\end{align*} sides, this means that each exterior angle is \begin{align*}\frac{360}{n}^\circ\end{align*}.

Example A

What is \begin{align*}y\end{align*}?

\begin{align*}y\end{align*} is an exterior angle and all the given angles add up to \begin{align*}360^\circ\end{align*}. Set up an equation.

\begin{align*}70^\circ + 60^\circ + 65^\circ + 40^\circ + y & = 360^\circ\\ y & = 125^\circ\end{align*}

Example B

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

Because the polygon is regular, the interior angles are equal. It also means the exterior angles are equal. \begin{align*}\frac{360^\circ}{7} \approx 51.43^\circ\end{align*}

Example C

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 \begin{align*}360^\circ\end{align*}.

CK-12 Exterior Angles in Convex Polygons

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Guided Practice

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

1. 12-gon

2. 100-gon

3. 36-gon

Answers:

For each, divide \begin{align*}360^\circ\end{align*} by the given number of sides.

1. \begin{align*}30^\circ\end{align*}

2. \begin{align*}3.6^\circ\end{align*}

3. \begin{align*}10^\circ\end{align*}

Explore More

  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 \begin{align*}x^\circ, 2x^\circ, 3x^\circ,\end{align*} and \begin{align*}4x^\circ.\end{align*} What is \begin{align*}x\end{align*}?

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

  1. octagon
  2. nonagon
  3. triangle
  4. pentagon

Answers for Explore More Problems

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

Vocabulary

exterior angle

exterior angle

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

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

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