What the numbers mean
Miter angle is the saw setting for both ends of every segment: 180° divided by the number of segments. Twelve segments is 15°, sixteen is 11.25°.
Outer segment length is the long side, measured long point to long point, and inner segment length the short side. Set a stop block to the outer length and every segment comes out the same.
Segment width is the strip to rip before you cut the segments: the wall thickness plus the allowance on both sides.
Why not just divide the circumference?
Glued-up segments make a polygon, not a circle. For that polygon to turn down to the diameter you want, its flat sides have to reach the circle, so each segment's long side is the diameter times tan(180° ÷ segments). Dividing the circumference by the number of segments comes up short: 2.3% at 12 segments and 5.3% at 8, enough to leave a ring undersize. This calculator uses the polygon, as Wood Calc does.
The extra allowance is added outside the ring and taken off inside it, so there's wood to turn true. A quarter inch per side is typical; use less once your glue-ups come out round.
A worked example
A 10" ring in 12 segments with a 3/4" wall and 1/4" allowance: set the saw to 15°, rip a 1 1/4" strip, and cut each segment 2 13/16" (2.813") long point to long point. The short side comes out about 2 5/32".
More than one ring
A bowl is a stack of rings, each a little wider or narrower than the one below. The Segmented Bowl Designer works out up to four rings at once, draws the profile, warns you when a ring doesn't overlap the one below it enough, and gives you a cut list for the whole bowl. Before you mount it, check the Lathe Speed Calculator. Turning from a log instead? The Bowl Blank Calculator shows the biggest bowl it gives, and the Sphere Turning Calculator lays out a ball.
Making your first one? How to Make a Segmented Bowl takes you from the design to cutting, gluing and turning the rings.
Building a flat polygon instead, like a hexagon box, an octagon frame or a planter? The Miter Angle Calculator gives the same cuts with no allowance, measured across the flats.