The sum of the exterior angles of any polygon is \begin{align*}360^\circ\end{align*}. How is this possible?
Polygons
A polygon is a flat shape defined by straight lines. A polygon is usually classified by its number of sides, as shown in the table below.
Number of Sides |
Name of Polygon |
3 |
Triangle |
4 |
Quadrilateral |
5 |
Pentagon |
6 |
Hexagon |
7 |
Heptagon |
8 |
Octagon |
9 |
Nonagon (or enneagon) |
10 |
Decagon |
For a polygon with more than 10 sides, most people prefer to name it by its number of sides and the suffix “gon”. For example, a 40 sided polygon would be a “40-gon”.
A diagonal is a line segment that connects any two non-adjacent vertices of a polygon. A polygon is convex if all diagonals remain inside the polygon. Most polygons that you study in geometry will be convex. If a polygon is not convex then it is concave (or non-convex). The blue pentagon on the left is convex, while the pink quadrilateral on the right is concave.
A polygon is equilateral if all of its sides are the same length. A polygon is equiangular if all of its angles are the same measure. A polygon is regular if it is both equilateral and equiangular.
The sum of the measures of the three angles in a triangle is \begin{align*}180^\circ\end{align*}. You can use this fact to find the sum of the measures of the angles in any polygon.
The pentagon above has been divided into three triangles, and its interior angles have been marked. The sum of the measures of the angles of each triangle is \begin{align*}180^\circ\end{align*}. Therefore, the sum of the interior angles of the pentagon is \begin{align*}180^\circ \cdot 3=540^\circ\end{align*}.
In general, the sum of the interior angles of a polygon with \begin{align*}n\end{align*} sides is \begin{align*}180(n-2)^\circ\end{align*}.
If the polygon is regular (and thus equiangular), you can figure out the measure of each interior angle. Let's take a look at some example problems.
Naming Polygons
Name the regular polygon below and find the sum of its interior angles.
This regular polygon has 7 sides, and is therefore a heptagon. To find the sum of the interior angles, imagine dividing the polygon into triangles. There are 5 triangles, each with \begin{align*}180^\circ\end{align*}. Therefore, the sum of the interior angles is \begin{align*}180^\circ \cdot 5=900^\circ\end{align*}.
Measuring Angles
Find \begin{align*}m\angle G\end{align*} in the polygon from the previous problem.
Because this is a regular polygon, it is equiangular. This means that each of the seven interior angles has the same measure. The sum of the interior angles was \begin{align*}900^\circ\end{align*}. This means that each of the seven interior angles is \begin{align*}\frac{900^\circ}{7}\approx 128.6^\circ\end{align*}.
Examining the Relationship Between Angles
An exterior angle is the angle between one side of a polygon and the extension of an adjacent side. In the polygon below, an exterior angle has been marked at vertex \begin{align*}G\end{align*}. How are exterior angles related to interior angles? What is the measure of the exterior angle at \begin{align*}G\end{align*}?
The exterior angle and interior angle at the same vertex will always be supplementary because together they form a straight angle. In this case, the interior angle at point \begin{align*}G\end{align*} was approximately \begin{align*}128.6^\circ\end{align*}. Therefore, the exterior angle is \begin{align*}180^\circ-128.6^\circ=51.4^\circ\end{align*}.
Examples
Example 1
Earlier, you were asked how it's possible that the sum of the exterior angles of any polygon is \begin{align*}360^\circ\end{align*}.
There are many ways to think about the sum of the exterior angles of a polygon. One way is to first consider that the sum of all the straight angles through the vertices is \begin{align*}180n^\circ\end{align*} (where \begin{align*}n\end{align*} is the number of sides of the polygon). These angles are marked in green in the sample polygon below.
If you only want the sum of the exterior angles, you must subtract the sum of the interior angles. The sum of the interior angles is \begin{align*}180(n-2)^\circ=180n^\circ-360^\circ\end{align*}. Therefore, the sum of the exterior angles is:
\begin{align*}180n^\circ-(180n^\circ-360^\circ)=180n^\circ-180n^\circ+360^\circ=360^\circ\end{align*}
Example 2
Find the sum of the interior angles of a nonagon.
A nonagon can be split into 7 triangles. The sum of the interior angles is \begin{align*}180^\circ \cdot 7=1260^\circ\end{align*}.
Example 3
Find the measure of one interior angle of a regular nonagon.
The sum of the interior angles is \begin{align*}1260^\circ\end{align*}. Therefore, each interior angle is \begin{align*}\frac{1260^\circ}{9}=140^\circ\end{align*}.
Example 4
Find the measure of one exterior angle of a regular decagon.
\begin{align*}\frac{360^\circ}{10}=36^\circ\end{align*}
Review
1. What is the measure of an exterior angle of a regular 45-gon?
2. What is the sum of the interior angles of a 35-gon?
3. Draw an example of a convex polygon and a concave polygon.
4. What's the name of a polygon with 8 sides?
5. What's the name of a polygon with 10 sides?
6. What's the name of a polygon with 4 sides?
7. How could you use the dissection shown in the picture below to show why the sum of the interior angles of a hexagon is \begin{align*}720^\circ\end{align*}?
8. How could you use the dissection shown in the picture below to show why the sum of the interior angles of a hexagon is \begin{align*}720^\circ\end{align*}?
9. A regular polygon has an interior angle of \begin{align*}150^\circ\end{align*}. How many sides does the polygon have?
10. How could you use exterior angles to help you find the answer to #9?
11. What is the sum of the exterior angles of an 11-gon?
12. What is the sum of the interior angles of an 11-gon?
13. Solve for \begin{align*}x\end{align*}:
14. Solve for \begin{align*}x\end{align*}:
15. Solve for \begin{align*}x\end{align*}:
Review (Answers)
To see the Review answers, open this PDF file and look for section 1.3.