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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

enter image description here

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

2 Answers 2

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If you're asking for an expression for the angles in a regular polygon, then here you are:

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 $$ \left(180 -\frac{360}{n}\right)^\circ $$

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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. $(1-\frac2n)\cdot 180^\circ$.

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