TITLE: How to calculate the radius of a circle composed of many smaller circles?
QUESTION [4 upvotes]: I'm making a CAD model of a ball bearing.
It contains 14 balls (perfect spheres) arranged in a perfect circle pattern.
So if each ball has a radius of $r$, how do I calculate the radius of the entire circular pattern?
(I'm looking for the radius of the circle containing the CENTER POINTS of all the balls.)
This example photo is a ball bearing with 9 balls, but mine has 14 balls - but this should help illustrate the situation:

REPLY [4 votes]: Imagine a line drawn from the center of each sphere to that of the next. That will product a $14$-sided regular polygon with side length $2r$. Its circumradius will be of length
$$R=\frac{r}{\sin\left(\frac{\pi}{14}\right)},$$
from which we conclude that the radius of the outer circle will be of length
$$r+R=r\left(1+\frac{1}{\sin\left(\frac{\pi}{14}\right)}\right).$$
(I'm not quite sure which of these two measurements you are interested in, so I provided both...)
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