Don't ask me how I worked this out, it took me a few hours. There are times when I am mathematically challenged Perhaps Marv will jump in here and explain the finer points
When people are faced with problems that involve arranging small circles around a large circle they often want to divide the circumference of the large circle by the diameter of the smaller circles to determine the spacing. That is dead wrong. Take a few minutes with pencil and paper to convince yourself that it won't work.
The way to approach these problems is to find the angle that the small circle subtends at the center of the larger circle. I'll let it as an exercise for the student to show that, if
r = radius of small circle
R = radius of large circle
the angle subtended at the center of the large circle is given by
A = 2 * arcsin (r/R)
Let's see how this would work for Rich's parameters. With r = 0.25 (0.5" diameter circles), how big must R be to accommodate nine small circles?
A = 360/9 = 40 deg so
arcsin (0.25/R) = A/2 = 20 deg
Taking the sine of both sides of the equation, we have
0.25/R = sin (20) = 0.34
So
R = 0.25/0.34 = 0.735
and the enclosing circle would have a radius of
R + r = 0.735 + 0.25 = 0.985
IOW, very close to the two inch diameter Rich found by his manipulations.
If you're dealing with problems of this sort (eg, laying out an elbow engine cylinder), you may find the REVOLVER program on my page helpful. While originally written to assist me in designing tool holders, it's general enough to be applicable to other forms.