How this instrument works
Segmented woodturning builds a bowl, vase or vessel out of many individual wedge-shaped pieces glued edge to edge into a ring, rather than turning it from a single solid blank. Each ring is a regular polygon — a shape with equal sides and equal angles — approximating a circle, and the more segments it has, the closer that polygon gets to looking round once it's on the lathe.
Two numbers define the cut for every segment in a ring: the miter angle, which is the angle each segment's edge must be cut to so the pieces meet cleanly all the way around, and the segment length, the straight-line (chord) length of each individual piece measured along the outer diameter. The miter angle comes from splitting a half-circle, 180°, evenly across the segment count, since each segment's two mitered edges together account for one full corner of the polygon. The segment length comes from the same trigonometry used for any regular polygon inscribed in a circle: outer diameter multiplied by the sine of the miter angle.
More segments make a smoother, rounder-looking ring but each cut angle gets closer to a straight line, cutting a piece with a shallow, error-prone miter — a common tradeoff segmented turners weigh between visual smoothness and how forgiving the joinery is to cut accurately. Fewer segments make an obviously faceted ring with a more generous, easier-to-cut miter angle, which is why beginners often start with 6 or 8 segments per ring before working up to 12, 16 or more.
- Enter the ring's segment count into Number of segments — the instrument needs at least 3 to define a polygon.
- Enter the ring's outer diameter into Outer diameter (in), measured across the widest point of the finished ring.
- Segment height (in, informational) records the ring's thickness for your own reference; it does not change the miter angle or segment length, since both depend only on the diameter and segment count.
- Read Miter angle per segment and set your table saw or miter gauge to that angle before cutting.
- Read Segment (chord) length for the straight-line length to cut each individual segment piece to.
Worked example — an 8-segment ring, 10-inch outer diameter
Enter 8 into Number of segments and 10 into Outer diameter (in); Segment height (in, informational) can stay at any value since it doesn't affect these two outputs. Miter angle per segment reads 22.5° — 180° split eight ways — the angle to set on the miter gauge or table saw blade for every one of the eight pieces.
Segment (chord) length reads 3.8268 in, from 10 × sin(22.5°). Cut eight pieces to that length with both ends mitered at 22.5°, glue them edge to edge, and the ring closes into a regular octagon whose corner points sit on a 10-inch circle — the outer diameter entered above.
Questions
Why is the miter angle 180° divided by the segment count, not 360°?
Because each segment contributes one corner to the ring, and each corner is formed by two mitered edges meeting — one from the segment on either side. Splitting a full 360° polygon into n corners means each corner's total angle is 360°/n, and since two identical miter cuts share that corner equally, each individual cut is half of that: 180°/n. For 8 segments that's 22.5° per cut, not 45°.
Does segment height affect the miter angle or segment length?
No — segment height is recorded purely for your own reference and doesn't enter the formula for either output. Miter angle depends only on how many segments make up the ring, and segment length depends only on the segment count and the outer diameter. A taller or shorter ring of rough-cut stock uses the identical miter angle and segment length as long as the segment count and diameter stay the same.
How many segments should a beginner start with?
6 or 8 segments is a common starting point, since the resulting miter angle (30° or 22.5°) is generous enough to cut accurately on a standard miter gauge, and small cumulative errors are less likely to leave a visible gap when the ring is glued up. Rings with 16, 24 or more segments look rounder but the miter angle shrinks toward a shallow cut, where the same small error in blade angle multiplies into a bigger visible gap around the full ring.
Why does a 6-segment ring's chord length equal exactly the radius?
Because a regular hexagon inscribed in a circle is made of six equilateral triangles meeting at the center — a classical geometric result — so each side length equals the circle's radius exactly. Enter 6 segments and any outer diameter, and Segment (chord) length will always come out to exactly half the entered diameter, which is a handy mental check on the instrument's output.
Is the segment length the same as the board width I need to rip?
Segment length is the outer chord length of the finished piece, not the raw board width before cutting the miters — you'll typically rip stock somewhat wider than this figure to leave material for the mitered ends and any final sanding or trimming. Treat the instrument's output as the target finished dimension along the outer diameter, then work backward to the blank size your particular joinery method needs.
Does this formula work for a ring made of a different shape, like a rectangle stack?
This instrument assumes a ring built from identical wedge-shaped segments arranged in a single circular layer — the standard segmented-turning approach. Stacked-ring or coopered constructions built from other segment shapes follow different layout math and aren't what this formula computes.