trigonometry
I am trying to determine how to calculate the required radius of the smaller circles so they touch each other around the larger circle. (red box)
I would like to be able to adjust the number of smaller circles and the radius of the larger circle.
As an example:
$$\begin{align}
R&=1.5\\
n&=9\\
r&=\,?
\end{align}$$
Here's a picture of the case $n=5$. 
If $R$ is the radius of the big circle and $r$ the radii of the small circles, do you see that the isosceles triangle has two sides $R-r$ and one side $2r$?
Let $\theta$ be the inscribed angle corresponding to the arc in radians, then we know:
$$R\theta=3.7855$$
$$(R+3.048) \theta=13.4$$
This is a system of two equations in two unknowns, which is easy to solve.
Best Answer
If you draw $n$ lines from the origin touching the small circles and $n$ lines from the origin to the center of each small circle you basically divide the $2 \pi$ angle into $2n$ equal angles, say $\theta$. Hence $\theta={\pi}/{n}$. Now a triangle with vertices the origin, the center of one of the small circles and a tangent point of the same circle is a right triangle since the tangent is perpendicular to the radius at the point of contact. You then have
$$ \sin \theta=\frac{r}{R} $$
Putting it all together
$$ r=R \sin \frac{\pi}{n} $$