How do I calculate the diameter of a chainring from the number of teeth? Simply knowing the number of teeth on a chainring can we determine an accurate diameter?
A chainring is an n-sided regular polygon where n is the number of teeth.  The side length s of the polygon is the distance from tip-to-tip of each chainring tooth.
The formula for radius of a regular polygon is:

(source: mathopenref.com)
Using zenbike's 12.75mm above for s, we get 107.61 for the radius, or 215.22mm for the diameter, which is very close to his approximation.
Comparing the two formulas shows that the length term, as expected, can be eliminated. This leaves us with:
1 / sin( pi / n ) vs. n / pi
For large n, those terms converge, introducing an error of just .12mm when n=53. It's a bit larger as n gets smaller, differing by .64mm for n=11.
For all practical purposes, I'd just use s * n / pi, even for the smallest cog you'll come across it'll be within a millimeter.