In what should be an easy to find formula, I've wasted my time searching for a relationship among the radius, chord, and arc length of a circle and yet all I come across are intermediate conversions to get to angles and then to what I want. We should be able to bypass the angle to simplify the process.

Letting L=arc length
r=radius
c=chord length

L=2r*arcsin(c/(2r))


Similarly, if knowing just the radius and segment height (h),

L=2r*arctan(sqrt(r^2-(r-h)^2)/(r-h)) = 2r*arctan(sqrt(2rh-h^2)/(r-h))


Updated 12/13/2014