Every ring in a segmented bowl needs its own segment length, and the miter has to be exact or the ring will not close. Give the diameter and the number of segments and this returns the cut angle, the blank size and how much strip to prepare.
Notes
Why the angle must be exact
With 12 segments each joint has two 15 degree cuts, so 24 cuts have to sum to 360 degrees. A tenth of a degree of error becomes a 2.4 degree gap by the time the ring closes — visible, and it will not glue up tight.
Cut to the outside, size to the inside
The segment length given here is the long side, on the outside of the ring. The short side is what determines whether the ring closes on the inside — both are given so you can check your blanks before gluing.
Ring width and waste
A ring turns down considerably: you lose the corners of every segment when you round it. The blank width here is the flat width before turning, and 15 to 20 percent extra strip is normal.
What this does not do
It sizes one ring at a time. Stacking, orientation of the grain and the wall thickness of the finished vessel are design decisions, not calculations.