What if you were given a twelve-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.
Watch This
CK-12 Foundation: Chapter6ExteriorAnglesinConvexPolygonsA
Watch the second half of this video.
James Sousa: Angles of Convex Polygons
Guidance
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 \begin{align*}360^\circ\end{align*}. Let’s extend this theorem to all polygons.
Investigation: Exterior Angle Tear-Up
Tools Needed: pencil, paper, colored pencils, scissors
- Draw a hexagon like the hexagons above. Color in the exterior angles as well.
- Cut out each exterior angle and label them 1-6.
- 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 \begin{align*}360^\circ\end{align*}, 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 \begin{align*}360^\circ\end{align*}.
Proof of the Exterior Angle Sum Theorem:
Given: Any \begin{align*}n-\end{align*}gon with \begin{align*}n\end{align*} sides, \begin{align*}n\end{align*} interior angles and \begin{align*}n\end{align*} exterior angles.
Prove: \begin{align*}n\end{align*} exterior angles add up to \begin{align*}360^\circ\end{align*}
NOTE: The interior angles are \begin{align*}x_1, x_2, \ldots x_n\end{align*}.
The exterior angles are \begin{align*}y_1, y_2, \ldots y_n\end{align*}.
Statement | Reason |
---|---|
1. Any \begin{align*}n-\end{align*}gon with \begin{align*}n\end{align*} sides, \begin{align*}n\end{align*} interior angles and \begin{align*}n\end{align*} exterior angles. | Given |
2. \begin{align*}x_n^\circ\end{align*} and \begin{align*}y_n^\circ\end{align*} are a linear pair | Definition of a linear pair |
3. \begin{align*}x_n^\circ\end{align*} and \begin{align*}y_n^\circ\end{align*} are supplementary | Linear Pair Postulate |
4. \begin{align*}x_n^\circ+ y_n^\circ=180^\circ\end{align*} | Definition of supplementary angles |
5. \begin{align*}(x_1^\circ+x_2^\circ+\ldots+x_n^\circ)+(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)=180^\circ n\end{align*} | Sum of all interior and exterior angles in an \begin{align*}n-\end{align*}gon |
6. \begin{align*}(n-2)180^\circ=(x_1^\circ+ x_2^\circ+\ldots+x_n^\circ)\end{align*} | Polygon Sum Formula |
7. \begin{align*}180^\circ n=(n-2)180^\circ+(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)\end{align*} | Substitution PoE |
8. \begin{align*}180^\circ n=180^\circ n-360^\circ+(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)\end{align*} | Distributive PoE |
9. \begin{align*}360^\circ=(y_1^\circ+ y_2^\circ+\ldots+ y_n^\circ)\end{align*} | Subtraction PoE |
Example A
What is \begin{align*}y\end{align*}?
\begin{align*}y\end{align*} is an exterior angle, as well as all the other given angle measures. Exterior angles add up to \begin{align*}360^\circ\end{align*}, so 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, each interior angle is equal. This also means that all the exterior angles are equal. The exterior angles add up to \begin{align*}360^\circ\end{align*}, so each angle is \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*}.
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter6ExteriorAnglesinConvexPolygonsB
Concept Problem Revisited
The exterior angles of a regular polygon sum to \begin{align*}360^\circ\end{align*}. The measure of each exterior angle in a dodecagon (twelve-sided regular polygon) is \begin{align*}\frac{360^\circ}{12} = 30^\circ\end{align*}.
Vocabulary
An exterior angle is an angle that is formed by extending a side of the polygon. A regular polygon is a polygon in which all of its sides and all of its angles are congruent.
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*}
Practice
- 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 \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:
- octagon
- nonagon
- triangle
- pentagon
- 50-gon
- heptagon
- 34-gon
- 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 \begin{align*}\frac{360^\circ}{n}\end{align*} and the other is \begin{align*}180^\circ - \frac{(n-2)180^\circ}{n}\end{align*} (\begin{align*}180^\circ\end{align*} minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent.
- Angle Puzzle Find the measures of the lettered angles below given that \begin{align*}m \ || \ n\end{align*}.