Given n circles of radius r, and one circle C of radius R. What is the maximum radius, MAX_R, of C for which all circles of radius r can sit around C and touch. Note: D >= d
I want to know the maximum radius for C and also find a function for R depending on n.
If all the outer circles are arranged perfectly around and touching $C$ then their centers are arranged in a regular polygon with $n$ sides. Each side of the polygon is equal to $2r$.
Take angle $\angle XYZ$ where
$X$ is the center of $C$,
$Y$ is the center of a circle c and
$Z$ is the center of a neighbouring circle.
We know that $\angle YXZ = 360 / n$, and $\triangle XYZ$ is an isosceles triangle so $\angle XYZ = \frac{180 - (360/n)}{2}$
Split $\triangle XYZ$ into a two identical right-angle triangles with base $r$ and work out that $XY = r / cos(\angle XYZ )$
The radius of $C$: $R = XY - r$
The full equation $R = r / cos(90 - (180/n)) - r$