The formula for pitch circle diameter.

Question

I want to put $n$ number of circle with $r$ radius each in a big circle. Want to calculate the radius $R$ of the big circle. How can this be achieved?

Answer 1

Consider a regular $n$-gon centered at the origin and with side length $2r$. Then the vertices are the centers of the $n$ small circles, and the distance $\rho$ from the center to a vertex is given by the relationship $\sin \frac{\pi}{n} = \frac{r}{\rho}$. Hence $$R = \rho + r = r\left( 1 + \csc \frac{\pi}{n} \right).$$