Apologies for the title, I found it hard to express this succinctly in words alone!
Consider the following image, where A is a known angle (<=180°) and d is the known diameter of a circle. Is it possible to calculate the smallest possible distance (x) between the centre of the circle and the corner of the "wedge" o?
It's $d/(2 sin(A/2))$. Draw a radius from the center of the circle to the point of tangency, and you have a right triangle with the angle $A/2$ opposite a side of length $d/2$, and a hypotenuse of length $x$. Solve $(d/2)/x = sin(A/2)$ for $x$ and you have your length.