Calculating the turning radius of a robot

Hi Everyone,
I have been working on a new wall smoothing algorithm and I've run into a 
bit of an issue I cant seem to figure out.  I'm trying to calculate the 
turning radius of a robot given its velocity.  I can get its change in 
degrees per tick based on its velocity *10 - 0.75 * Math.abs(getVelocity())*, 
but that doesn't really help me find the radius of its arc.

By plotting the points of my bot along its arc I was able to estimate its 
radius.  These values are close, but not 100% accurate. Does anyone know of 
a proper formula for calculating the radius without requiring 3 sample 
points along the arc?

Here are my findings: Velocity : Arc Radius

8 : 114.627
7 :  84.464
6 : 62:159
5: 45.862
4: 32.764
3: 22.186
2: 13.492
1: 6.205
0 : 0

Thanks for your time!

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