Not quite clear what you are asking yet, but here’s a helpful octagon calculator.
You want a circle inscribed in the octagon where the circle radius is 2" less than the octagon radius (center to midpoint of a side), is that correct? I think I’m getting what you are asking about the sides. Before I screw up the math trying to figure this out (@naita sounds right to me) I’ll suggest you not stick to the specific 2" and pick your lumber for a close match instead of a precise number.
And I’m sure you know you need to leave some extra space around the glass circle to allow for some expansion and contraction of both materials.
And finally, I’d suggest waiting a while before buying any lumber, we’re at the top of a price bubble right now.