The calculation
Define the diameter before using a formula
A polygon has a larger diameter across its corners than across its flats. This tool defines the input as twice the outer apothem: the perpendicular distance from the center to a flat. For an odd-sided polygon, this is the diameter of its inscribed circle, not a distance between two parallel opposite sides.
If you start from an across-corners measurement, use a matching conversion first. Mixing conventions changes the segment length and can leave too little stock for the intended turned shape.
Calculate the trapezoid
Let D be the outside across-flats diameter, w the radial strip width, and n the number of segments. The long edge is D × tan(π/n). The short edge is (D − 2w) × tan(π/n). Each end is cut at a saw miter of 180°/n away from square. Across-corners diameter is D ÷ cos(π/n).
For D = 12 in, w = 1.5 in and n = 12, the long edge is approximately 3.2154 in, the short edge 2.4115 in, and each miter 15°. The outer corners span about 12.4233 in. The diagram uses those same geometric definitions.
This is a ring, not a complete bowl design
The tool solves one flat, regular segmented ring. It does not design a vessel profile, compound stave angle, glue joint, wood movement allowance, wall thickness after turning or safe turning procedure. Stack design and lathe work require additional planning.
The sum of long edges is shown only as a geometric reference. It is not a minimum purchase length: kerf, angle-cut layout, alternating stock orientation, end trim and defects can all change the strip length required. Make a test ring before cutting the full set.
Questions from the workbench
What if I want a 12-inch turned outside diameter?
The outer across-flats input must include any turning and truing allowance you need. A mathematically exact 12-inch blank leaves no allowance at its flats.
Can this calculate a stave vessel or compound miter?
No. It assumes a flat ring with vertical sides and no bevel tilt. Use a separate, validated geometry for compound constructions.
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