Bowl Segment Calculator (segmented woodturning ring: miter angle & segment lengths)

Understanding the bowl segment calculator — concept and principle

This calculator works out the miter angle and segment lengths for a segmented woodturning ring, and the board width and length needed to cut one ring. Woodturners use it to plan segmented bowls and vessels.

Ring of equal segments: with the default inputs the result is 15° miter (board 23.8 × 622 mm).

How it is calculated

Segment angle = 360° ÷ N; miter angle on each end = 180° ÷ N

Outer radius R′ = outside diameter ÷ 2 + allowance; inner radius r′ = outside diameter ÷ 2 − ring width − allowance

Outside edge = 2 × R′ × tan(180° ÷ N)

Inside edge = 2 × r′ × cos(180° ÷ N) × tan(180° ÷ N)

Board width = R′ − r′ × cos(180° ÷ N)

Board length for one ring ≈ N × (average edge length + kerf)

The principle

A ring of N segments is a regular N-sided polygon. Each segment spans 360° ÷ N, so each end is cut at half of that from square. To leave wood for turning, the polygon must surround the outside circle and its inner corners must stay inside the inner circle. Trigonometry of regular polygons links the radii to the edge lengths. Segments are cut alternately from a strip, so each uses about the average edge plus one saw kerf.

Worked example

Default inputs: 12 segments, outside diameter 200 mm, ring width 15 mm, allowance 3 mm, kerf 3 mm.

Segment angle = 360 ÷ 12 = 30°; miter = 15°.
R′ = 103 mm; outside edge = 2 × 103 × tan 15° = 55.2 mm.
r′ = 82 mm; inside edge = 2 × 82 × cos 15° × tan 15° = 42.4 mm.
Board width = 103 − 82 × cos 15° = 23.8 mm; board length ≈ 12 × (48.8 + 3) = 622 mm.

Each point is the calculator re-run with only “Segments in the ring” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Segments in the ring	Miter angle on each end (°)
3	60
6	30
9	20
12	15
15	12
18	10
24	7.5

How to read the result

Small angle errors add up around the ring: an error of 0.1° per cut becomes 2.4° over 12 segments with 24 cuts. Make test cuts and adjust the saw, or sand joints flat on a disc. Add extra length for setup.

Common mistakes

Setting the saw to the segment angle (30°) instead of the miter angle (15°).
Forgetting the turning allowance, leaving no wood to shape.
Ignoring the kerf when estimating board length.

Keep in mind: Use guards and push sticks when cutting small segments, and wear eye protection on the lathe.

Frequently asked questions

What miter angle for 12 segments?

15° on each end.

What about 8 segments?

22.5° on each end.

Why add an allowance?

So there is wood to turn away to a smooth curve inside and out.

Sources: Plane geometry of regular polygons (central angle 360°/N, apothem R·cos(180°/N), side 2·a·tan(180°/N)); e.g. Zwillinger D (ed.), CRC Standard Mathematical Tables and Formulas, 33rd ed., 2018

Checked 9 Nov 2026. Plane geometry of regular polygons: central angle 360°/N, apothem a = R·cos(180°/N), side = 2·a·tan(180°/N) (circumscribed and inscribed circles).

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