A segmented bowl is a stack of rings, and each ring is a regular polygon made from wedges glued edge to edge. Everything about cutting it comes from one piece of geometry: the angle at which you cut the ends of each wedge, and how long the wedge has to be so that the finished polygon still contains the circle you intend to turn. This segmented bowl calculator gives you both, plus the stock width and the board footage for the whole vessel.
Arb Digital publishes free calculators across construction, woodworking and workshop trades. This one exists because the arithmetic is genuinely fiddly — the outer and inner edges of a segment are different lengths, the blank has to be oversize in two directions at once, and the kerf you lose at every cut adds up fast when a single ring takes twenty-four cuts.
What This Segmented Bowl Calculator Does
Enter how many segments you want in a ring, the finished outside diameter, the finished wall thickness and the thickness of the stock you are cutting from, and it returns the miter angle for each end, the long outer edge and short inner edge of every segment, the board width you need to cut the strips at, and the total board feet for the whole bowl including your allowances.
The two breakdown bars compare the outer and inner edges of a segment, which is a useful visual check: on a thin-walled ring the two lengths are close, and on a thick-walled ring the wedge becomes noticeably tapered. If the inner edge ever comes out as zero or negative, the wall thickness you asked for is more than the radius allows and the calculator says so instead of printing nonsense.
What it does not do is design the vessel. Wood movement, grain orientation, glue choice, feature ring layout and how a stack behaves as it seasons are craft decisions, not arithmetic, and a calculator has nothing useful to say about them.
How to Use It
- Pick the segment count. More segments give a rounder blank and less waste, but more joints and more chances for cumulative angle error. Twelve and sixteen are the usual compromise.
- Enter the finished outside diameter and wall thickness. These are the dimensions of the turned ring, not the blank.
- Set the ring height. This is the thickness of the board the segments are cut from and it becomes the height of one course of the bowl.
- Add the turning allowance and the kerf. The allowance widens the strip so there is material to true away inside and out; the kerf is lost twice per segment.
- Set the number of rings and a stock allowance to get the total board footage for the whole vessel.
The Formula and How It Is Calculated
A ring of n segments is a regular n-sided polygon. The interior angle at each corner is 180 × (n − 2) ÷ n, and since each corner is formed by two cut ends meeting, the cut on each end is half the exterior angle: miter angle = 180 ÷ n degrees. Twelve segments gives 15°, sixteen gives 11.25°, eight gives 22.5°. That is the number you set on the saw, measured from square.
Segment length follows from the polygon's apothem, the perpendicular distance from the centre to the middle of a side. For the finished ring to contain a circle of radius R, the polygon must be circumscribed about that circle, so the apothem equals R and the side length is 2 × R × tan(180° ÷ n). That gives the outer edge. The inner edge uses the inner radius, R minus the wall thickness, on the same radial lines: 2 × (R − t) × tan(180° ÷ n). The relationships between apothem, side and radius for a regular polygon are set out in the Wolfram MathWorld entry on regular polygons if you want the derivation.
Board width is the wall thickness plus your turning allowance. Strip length per ring is n × the outer edge, plus 2n × the kerf, because every segment costs two cuts. Board feet is thickness × width × length in inches divided by 144, multiplied by the number of rings and the stock allowance.
Worked example you can check: twelve segments, 10 inch outside diameter, ¾ inch wall, 1 inch stock, ⅛ inch kerf, ¼ inch allowance, six rings. The miter angle is 180 ÷ 12 = 15°, and tan 15° = 0.26795. The outer edge is 10 × 0.26795 = 2.680 inches; the inner edge is (10 − 1.5) × 0.26795 = 2.278 inches. Board width is 0.75 + 0.25 = 1.00 inch. Strip per ring is 12 × 2.680 + 24 × 0.125 = 32.15 + 3.00 = 35.15 inches, which at 1 inch by 1 inch is 35.15 ÷ 144 = 0.244 board feet. Six rings is 1.465 board feet, and with a twenty per cent allowance, 1.758.
Why the Polygon Is Circumscribed, Not Inscribed
This is the mistake that ruins a first bowl. If you compute the side length of a polygon inscribed in the finished circle — the formula with sine in it that most polygon references give first — the flats of the polygon fall inside the intended diameter, and when you turn the ring round you cut straight through the middle of every segment. The blank has to be big enough that the finished circle fits inside the flats, which means the apothem, not the circumradius, equals the finished radius.
The practical consequence is that the corners of the ring stick out beyond the finished diameter by a predictable amount: the circumradius of the polygon is R ÷ cos(180° ÷ n). On a twelve-segment 10 inch ring that is 5 ÷ cos15° = 5.176 inches, so the glued-up ring measures 10.35 inches across the points before turning. Knowing that number tells you whether the blank will clear your lathe bed. Our polygon calculator covers the general apothem, circumradius and side relationships if you want to explore other segment counts.
Cumulative Error and Why Test Rings Exist
The unforgiving thing about segmented work is that angle errors accumulate around the ring rather than cancelling. A twelve-segment ring has twenty-four cut faces, so a saw set one tenth of a degree fat closes the circle 2.4 degrees short and leaves a visible gap at the last joint. Sixteen segments has thirty-two faces and is correspondingly less forgiving of the same error.
Turners deal with this by cutting a full test ring in cheap stock, dry-clamping it, and reading the gap. A gap that opens on the outside means the angle is slightly too large; a gap on the inside means it is slightly too small. The correction is half the error divided by the number of joints, which is finer than most saw scales read — which is why sleds with a micro-adjustable fence stop are standard equipment. The miter angle calculator covers the general saw-setting problem for corners that are not part of a closed ring, including compound bevel work.
Choosing a Segment Count
More segments means a blank closer to round, so less material is turned away and less stock is wasted. It also means shorter, easier-to-handle pieces and more scope for alternating species around the ring. Against that, every extra segment adds two more cuts, two more glue joints and two more opportunities for the angle error described above, and short segments are fiddlier to hold safely at the saw.
Eight segments gives a 22.5° cut and a chunky, visibly faceted blank. Twelve at 15° is the common default. Sixteen at 11.25° and twenty-four at 7.5° are used for large diameters where the flats would otherwise be pronounced, and for feature rings where a fine repeat is the point. There is no correct answer; the calculator will run any count from three upwards so you can compare the numbers directly.
Stock, Waste and Planning the Cut List
Segmented work is unusually wasteful because you are cutting wedges from rectangular strips and throwing away the triangles. The board-foot figure here counts the strip you consume, not the finished wood in the bowl, which is the number you actually need when buying. A twenty per cent allowance on top is reasonable for a first attempt and can come down with experience.
If you are pricing timber, the board foot calculator converts dimensions and quantity into board feet and cost at your own supplier's rate, and the plywood sheet calculator covers sheet goods if you are building the jigs. The cord of wood calculator is for firewood volume rather than lumber, which is a different measure entirely. General woodturning practice, safety and technique are well covered by the American Association of Woodturners, which publishes free safety material for turners.
Arb Digital publishes free calculators for woodworking, geometry, materials and site take-offs. Browse the library, or tell us what is missing.
Browse the free tools hub Contact Arb DigitalCommon Mistakes to Avoid
- Using the inscribed polygon side length. It makes every segment too short, and the finished turning cuts through the flats. The apothem must equal the finished radius, not the circumradius.
- Cutting to the finished wall thickness exactly. With no radial allowance there is nothing to true away, and any ring that glues up slightly out of round cannot be recovered.
- Ignoring kerf on the strip length. Two cuts per segment on a sixteen-segment ring is thirty-two kerfs, which on a thick blade is four inches of stock per ring.
- Trusting the saw scale. The scale resolution is coarser than the accuracy a closed ring demands. Cut a test ring and adjust from the measured gap.
- Mixing species without thinking about movement. Woods with different shrinkage behaviour glued into a rigid ring will stress the joints as the moisture content changes. That is a craft judgement this page cannot make for you.
Related Free Tools From Arb Digital
Set the saw for corners and compound cuts with the miter angle calculator, explore apothem and circumradius relationships with the polygon calculator or the circle calculator, price timber with the board foot calculator, and lay out sheet goods with the plywood sheet calculator. Firewood is measured differently — see the cord of wood calculator. Everything else is in the free tools hub.
Frequently Asked Questions
Fifteen degrees on each end, because the miter angle is 180 divided by the number of segments. Sixteen segments gives 11.25 degrees and eight gives 22.5 degrees. Each joint is formed by two cut faces, which is why the angle is half the exterior angle rather than the whole of it.
The outer edge is the finished diameter multiplied by the tangent of 180 divided by the segment count. For a 10 inch ring in twelve segments that is 10 × tan 15° = 2.68 inches. The inner edge uses the inner diameter on the same formula.
Because the corners of a polygon stick out past its flats. The distance across the points is the finished radius divided by the cosine of 180 divided by the segment count, doubled — about 10.35 inches for a 10 inch twelve-segment ring. That is normal and it is turned away.
Far more than the finished wood in the bowl, because you cut wedges from rectangular strips. The board-foot figure here counts the strip consumed including kerf and your allowance, which is the quantity to buy rather than the quantity that ends up in the vessel.
It gives a rounder blank and less turning waste, but it adds joints, cuts and cumulative angle error, and short segments are harder to handle safely. Twelve and sixteen are the common compromise; the calculator will run any count so you can compare.
Almost always a small angle error multiplied around the ring. A gap that opens on the outside means the cut angle is slightly large; a gap on the inside means it is slightly small. Correct by half the measured error spread across the joints, and cut a test ring before committing good stock.
Yes, on the strip length. Every segment costs two cuts, so the kerf is applied twice per segment when working out how much stock a ring consumes. It does not change the finished segment dimensions, which are what you cut to.
This tool computes segmented turning geometry and a stock estimate from the dimensions you enter. It is a planning aid. It makes no judgement about wood selection, grain orientation, adhesive, moisture movement or whether a given blank can be turned safely on your lathe — those remain the turner's decisions, made with the manufacturer's guidance for the equipment and materials in use.
