http://www.mathopenref.com/chord.html
The chord length of the diameter of the circle of radius R, on the sphere of radius r, is given by the formula
2*R = 2* math.sqrt(r*r - d*d)
R*R = r*r - d*d
d*d = r*r - R*R


So the center of the circle is located at a distance "d" from the center of the sphere, where

d = sqrt(r*r - R*R)

So a simple script may be possible, if the radius of the sphere and the radius of the circle, can be extracted by MoI.

d is the Apothem, also denoted by a.
The depth of the center of the circle, from the surface of the sphere, is the Sagitta. Which sounds a lot like "sag" :-)