A segmented ring is a circle made from straight pieces mitred at both ends. It is how round tabletops, curved aprons, bent-look rails and segmented bowl blanks are built without steam or laminating. The geometry is simple and the tolerance is brutal.
The two numbers you need
The miter is 180 divided by the number of segments — the same as any flat frame. Twelve segments is 15 degrees at each end.
The segment length comes from the radius. The outer edge of each segment is twice the outer radius times the tangent of 180 over the number of segments. The inner edge uses the inner radius the same way.
That difference between inner and outer edge length is the taper on each segment, and it is what makes the ring close into a circle rather than a straight run.
Why error multiplies
A twelve-segment ring has twenty-four mitred ends. If the saw is off by a tenth of a degree, the ring is off by 2.4 degrees by the time it closes, and that shows as a wedge-shaped gap at the last joint.
There is nowhere for the error to go. A rectangular frame can be pulled square with clamps; a ring cannot be pulled round.
So the order of work matters: set the saw, cut two test segments, and check the angle between them against a protractor or by stacking pairs. Only cut the real stock when a dry pair matches the target exactly.
Grain direction and the weak point
Each segment has grain running along its length, which means at every joint, long grain meets end grain at an angle. That is a weaker glue surface than a straight edge joint.
The standard fix is a second layer of segments glued on top with the joints staggered — offset by half a segment — so no joint runs through the full thickness. A two-layer segmented ring is dramatically stronger and is how most segmented tabletops are actually built.
It also doubles the stock requirement, so the figure this calculator gives is per layer.
Cut the blanks oversize in width too
The ring width entered here is the finished width. Cut the segments wider, glue the ring up, and then cut the circle — inside and outside — on a router trammel or a bandsaw with a circle jig.
Trying to cut each segment to its final tapered shape before glue-up means every segment has to be perfect in two dimensions instead of one. Shaping the ring after assembly means only the angle has to be right.
Half an inch extra on each edge is plenty, and the calculator adds a half inch to the blank length for the same reason.
Clamping a ring
Band clamps and ratchet straps are the usual answer, sometimes with a second strap in the opposite direction. Individual clamps at each joint fight each other and push the ring out of round.
A useful trick is to tape the joints on the outside of the ring with the segments laid flat and open, apply glue, then roll the ring closed — the tape acts as hinges and pulls each joint tight as it closes.
Work on a flat surface with wax paper under it, because a ring glued up out of plane is impossible to flatten afterwards without losing thickness.