When dealing with polygons in geometry, it sometimes helps to know the total measure of all interior angles. We can also find the exact measure of each angle in a regular polygon.
So, how many degrees are in a polygon? If we sum the interior angles of a polygon with N sides (an N-gon), there are 180(N-2) degrees. If the polygon is regular, then every interior angle has the same measure: 180(N-2)/n. The exterior angles of an N-sided polygon always sum to 360 degrees, regardless of the value of N.
Of course, for a polygon that is not regular, we would need to do a little more work to find the measure of an individual interior angle.
In this article, we’ll talk about how many degrees are in a polygon. We’ll also look at some specific examples for polygons from N = 3 to N = 10 sides.
Let’s get started.
How Many Degrees In A Polygon?
There are 180(N – 2) degrees in a polygon if we add up the measures of every interior angle:
- Sum of Interior Angles of an N-gon = 180(N – 2) degrees.
For example, a polygon with N = 22 sides has 180(22 – 2) = 180(20) = 3600 degrees. That is, the sum of all interior angles in a 22-sided polygon is 3600 degrees.
If we add up the measure of every exterior angle, we get a total of 360 degrees:
- Sum of Exterior Angles of an N-gon = 360 degrees.
The image below illustrates interior and exterior angles (in a hexagon).
Remember that for a regular polygon:
- every side has the same length
- every angle has the same measure
Given this second fact, we can easily calculate the measures of each interior and exterior angle for a regular polygon with N sides.
Since there are N angles of the same measure, we only need to divide the sum of the angles by N. Therefore:
- Measure of One Interior Angle of a Regular N-gon = 180(N – 2) / N degrees.
- Measure of One Exterior Angle of a Regular N-gon = 360 / N degrees.
Note what happens when we take the sum of an interior angle and an exterior angle for a regular N-gon:
- [Interior Angle of Regular N-gon] + [Exterior Angle of Regular N-gon]
- =[180(N – 2) / N] + [360 / N]
- =[(180N – 360) / N] + [360 / N]
- =(180N – 360 + 360) / N
- =180N / N
- =180
As expected, we get a sum of 180 degrees for the sum of an interior angle in a polygon and its corresponding exterior angle.
How Many Degrees In A Triangle?
There are 180 degrees in a triangle.
A triangle has three angles (or if you prefer, a trigon has 3 sides – this is where trigonometry comes from!). We can use the formula for the sum of interior angles to verify this:
- 180(N – 2)
- =180(3 – 2)
- =180(1)
- =180 degrees
For a regular 3-gon (that is, an equilateral triangle), the measure of each interior angle is:
- 180(N – 2) / N
- =180(3 – 2) / 3
- =60 degrees
As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 3-gon (equilateral triangle) is:
- 360 / N
- =360 / 3
- =120 degrees
How Many Degrees In A Quadrilateral, Rectangle, Or Square?
There are 360 degrees in a quadrilateral, rectangle, or square (that is, a 4-gon).
We can use the formula for the sum of interior angles to verify this:
- 180(N – 2)
- =180(4 – 2)
- =180(2)
- =360 degrees
For a regular 4-gon (that is, a square), the measure of each interior angle is:
- 180(N – 2) / N
- =180(4 – 2) / 4
- =180(2) / 4
- =90 degrees
As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 4-gon (square) is:
- 360 / N
- =360 / 4
- =90 degrees
Note: this is the only case where the interior and exterior angles for a regular N-gon have the same measure. Here is the proof:
- Measure of Interior Angle of Regular N-gon = Measure of Exterior Angle of Regular N-gon
- 180(N – 2) / N = 360 / N
- 180(N – 2) = 360
- 180N – 360 = 360
- 180N = 720
- N = 4
How Many Degrees In A Pentagon?
There are 540 degrees in a pentagon (that is, a 5-gon).
We can use the formula for the sum of interior angles to verify this:
- 180(N – 2)
- =180(5 – 2)
- =180(3)
- =540 degrees
For a regular 5-gon (that is, a regular pentagon), the measure of each interior angle is:
- 180(N – 2) / N
- =180(5 – 2) / 5
- =180(3) / 5
- =108 degrees
As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 5-gon (regular pentagon) is:
- 360 / N
- =360 / 5
- =72 degrees
How Many Degrees In A Hexagon?
There are 720 degrees in a hexagon (that is, a 6-gon).
We can use the formula for the sum of interior angles to verify this:
- 180(N – 2)
- =180(6 – 2)
- =180(4)
- =720 degrees
For a regular 6-gon (that is, a regular hexagon), the measure of each interior angle is:
- 180(N – 2) / N
- =180(6 – 2) / 6
- =180(4) / 6
- =120 degrees
As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 6-gon (regular hexagon) is:
- 360 / N
- =360 / 6
- =60 degrees
You can learn about uses of hexagons in real life (and where they appear in nature) here.
How Many Degrees In A Heptagon?
There are 900 degrees in a heptagon (that is, a 7-gon).
We can use the formula for the sum of interior angles to verify this:
- 180(N – 2)
- =180(7 – 2)
- =180(5)
- =900 degrees
For a regular 7-gon (that is, a regular heptagon), the measure of each interior angle is:
- 180(N – 2) / N
- =180(7 – 2) / 7
- =180(5) / 7
- =900 / 7
- ~128.57 degrees
As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 7-gon (regular heptagon) is:
- 360 / N
- =360 / 7
- ~51.43 degrees
How Many Degrees In An Octagon?
There are 1080 degrees in an octagon (that is, an 8-gon).
We can use the formula for the sum of interior angles to verify this:
- 180(N – 2)
- =180(8 – 2)
- =180(6)
- =1080 degrees
For a regular 8-gon (that is, a regular octagon), the measure of each interior angle is:
- 180(N – 2) / N
- =180(8 – 2) / 8
- =180(6) / 8
- =1080 / 8
- =135 degrees
As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 8-gon (regular octagon) is:
- 360 / N
- =360 / 8
- =45 degrees
How Many Degrees In A Nonagon?
There are 1260 degrees in a nonagon (that is, a 9-gon).
We can use the formula for the sum of interior angles to verify this:
- 180(N – 2)
- =180(9 – 2)
- =180(7)
- =1260 degrees
For a regular 9-gon (that is, a regular nonagon), the measure of each interior angle is:
- 180(N – 2) / N
- =180(9 – 2) / 9
- =180(7) / 9
- =1260 / 9
- =140 degrees
As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 9-gon (regular nonagon) is:
- 360 / N
- =360 / 9
- =40 degrees
How Many Degrees In A Decagon?
There are 1440 degrees in a decagon (that is, a 10-gon).
We can use the formula for the sum of interior angles to verify this:
- 180(N – 2)
- =180(10 – 2)
- =180(8)
- =1440 degrees
For a regular 10-gon (that is, a regular decagon), the measure of each interior angle is:
- 180(N – 2) / N
- =180(10 – 2) / 10
- =180(8) / 10
- =1440 / 10
- =144 degrees
As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 10-gon (regular decagon) is:
- 360 / N
- =360 / 10
- =36 degrees
The table below summarizes the interior angles and exterior angles for polygons with N sides, for N = 3 to 10.
| N | Interior Angle Sum | Interior Angle Measure (Regular N-gon) | Exterior Angle Measure (Regular N-gon) |
|---|---|---|---|
| 3 | 180 | 60 | 120 |
| 4 | 360 | 90 | 90 |
| 5 | 540 | 108 | 72 |
| 6 | 720 | 120 | 60 |
| 7 | 900 | ~128.57 | ~51.43 |
| 8 | 1080 | 135 | 45 |
| 9 | 1260 | 140 | 40 |
| 10 | 1440 | 144 | 36 |
angles and exterior angles for polygons
with N sides, for N = 3 to 10.
Conclusion
Now you know how many degrees are in a polygon (that is, the sum of the interior angles), depending on the number of sides. You also know how to find the measures of interior and exterior angles for regular polygons.
I hope you found this article helpful. If so, please share it with someone who can use the information.
Don’t forget to subscribe to my YouTube channel & get updates on new math videos!
~Jonathon