Figure 1: n-gon. Here . The figure was drawn using Geogebra with the following command: which produced the points. The segments were then added to join the points.
I want to derive a side length for a regular n-gon inscribed in a unit circle. So, starting with , I assigned the first point to and going counter clockwise, let the next 2 points be
and
respectively. After a few repetitions of this with other values of , I found that the vertices of this particular n-gon were at .
To get the side length, I use the distance formula and compute . That resulted in the following algebra:
Now use the identity to get
To get the side length when the radius is different from , we just need