# Dependence of internal angles on the number of sides of a regular polygon

Keeping things equilateral, the internal angles (in degrees) to the number of side goes thusly:

3   60
4   90
5   108
6   120
7   128.571
8   135
9   140
10  144
11  147.273

three being a triangle, four a square, and so on. I made a curve with it with a spreadsheet

• Try multiplying the individual angles by the number of angles and replot Sep 26, 2012 at 21:01
• The angles are $$\frac{\text{sides} - 2}{\text{sides}}\cdot 180^\circ$$ Sep 26, 2012 at 21:03
• Nice to remember as n goes to infinity, the degree approaches 180 if you're teaching kids.
– user307605
Jan 24, 2016 at 13:20

If you walk along the edge, all the way around, you will have turned a total of $360^\circ$, so in each corner, you turn $\frac{360^\circ}{n}$. The internal angle is the supplementary angle of this, and is therefore $$\left(180 -\frac{360}{n}\right)^\circ$$
Your regular $n$-gon can be cut into (non-regular) triangles by means of $n-2$ diagonals. Since the sum of internal angles in a triangle is $180^\circ$, the sum of internal angles in an $n$-gon is $(n-2)\cdot 180^\circ$ and the single angles are one $n$th thereof, i.e. $(1-\frac2n)\cdot 180^\circ$.