Vertices of a regular polygon given the incircle radius
what is the relation between radius of the in-circle and circum-circle of a polygon?


That is $cos ( \frac{2\pi}{n}*\frac{1}{2} ) = cos ( \frac{\pi}{n})$

The triangle with edges from the center to the middle of a side, from the center to an adjacent corner and half the side connecting them is a right angle triangle. The angle of the point at the center $ =\frac\pi 2$. 

Then it's simply the cosine to get the ratio between adjacent and hypotenuse.