Why ring diameters need two different boundaries
The outer flat-to-flat blank diameter is the largest round outside diameter a polygon can contain: turning larger would cut through a flat. Its corner-to-corner diameter is larger by 1 ÷ cos(180° / N). At the opening, the opposite boundary matters: inner corners must stay smaller than the final bore to leave inside turning allowance. This tool sets outer blank flats = finished OD + 2 × outside allowance, and inner opening corners = finished bore ID − 2 × inside allowance. It converts opening corners to flats using cos(180° / N). Staves and Segments distinguishes across-flat turnable diameter, across-points diameter, and ring opening.
Ring segment and stock formulas
For N equal trapezoids, a flat ring’s saw miter is 180° ÷ N from square crosscut; the blade remains square to the broad face. Outer and inner parallel edges are their across-flats blank diameters × tan(180° / N). Strip width is half the difference between those blank diameters. The American Association of Woodturners’ segmented-bowl guide gives the diameter × tangent relation and recommends planning each ring from a drawing.
The alternating nested strip starts with one long edge and advances by the average of long and short edges for each later piece. Its estimate adds N + 1 projected kerfs (kerf ÷ cos miter) and two end trims. Independent cutting budgets a long-edge rectangle and two projected end kerfs per piece. These are geometric stock estimates, not a safe cutting sequence, grain plan, stop-block setting, or guaranteed yield.
Tapered staves are another assembly
Tapered staves join edge to edge along their long sides; flat rings join end to end. For top and bottom across-flats outer diameters, each stave’s outer-face width is diameter × tan(180° / N). Slope from vertical follows radius change ÷ vertical height. With broad face laid flat, long-edge skew from stave length = atan(sin(slope) × tan(180° / N)); joint blade bevel from vertical = asin(cos(slope) × sin(180° / N)). The same geometry powers ToolQuill’s compound miter calculator. With zero slope, skew is zero and bevel equals 180° / N. This mode does not size wall thickness or a hollow bore. Use a suitable jig and verify assembly on scrap.
Frequently asked questions
Is the entered OD across flats or corners?
For rings, enter the desired finished turned outside diameter; the planner grows blank across-flats by your outside allowance. For staves, enter finished outside across-flats diameters at top and bottom.
Why is the inner opening across-flats diameter smaller than my bore?
The opening’s corner-to-corner diameter is held below the bore diameter by your inside allowance. With fewer sides, corners stand farther from flats, so the flats must move inward to keep every corner inside the final bore.
Can I use nested board length as a stop-block setting?
No. It is a theoretical strip envelope with modeled kerf and trims, not an individual segment stop length or proof of safe cut order. Use the outside and inside edge lengths to make a test ring; check blade kerf, alternating flips, angle accuracy, and your sled.
Does the stave bevel also apply to flat rings?
No. Flat end-mitered ring segments use 180° / N miter with a square blade. The stave bevel is for long edge-to-edge joints of a side assembly laid broad-face flat. Saw and stock orientation matter.
Does this produce a ready-to-turn bowl?
No. It plans geometric blanks only. Adhesive, clamp pressure, grain orientation, staggered joints, solid base, hollowing support, workholding, and final shape need a shop plan and test cuts.