The two numbers
Miter from square = 180° ÷ number of segments
Outer edge = outside diameter × tan(180° ÷ segments)
Four segments is a square, and the miter is 45°. Six is a hexagon, 30°. Eight is an octagon, 22.5°. Twelve, a common bowl ring, is 15°. The miter calculator is the rectangular case of the same angle. The segmented ring calculator adds the length of each side and the strip they come from.
The tangent length is the side of a polygon that reaches the finished circle, with the corners standing outside it. After the ring is glued, those corners are turned or planed away and the outside becomes the diameter you typed. A shorter piece, the plain chord, does not give you that.
Chord that falls short = outside diameter × sin(180° ÷ segments)
That sine length puts the corners on a smaller story. The middle of each segment sits inside the finished circle, and turning the corners off makes the ring smaller still. Cut the tangent length when the finished outside diameter is the circle you care about.
Wall, inside edge, and the strip
The finished inside diameter is the outside diameter minus twice the wall. The inside edge of each segment uses the same tangent on that smaller diameter. The wall you type is the finished wall. It does not include a turning allowance.
Inside diameter = outside diameter − 2 × wall
Stock length = segments × outer edge + (segments − 1) × kerf
One strip holds every outer edge in a row. There is a kerf between segments and no extra kerf after the last one. The angle does not change when you add kerf. Only the strip gets longer.
Worked examples
A 10 inch bowl ring, 12 segments, ½ inch finished wall, segments already cut so the kerf is zero. The miter is 180 ÷ 12 = 15°. The outer edge is 10 × tan(15°) = 2.679 inches. The plain chord is 10 × sin(15°) = 2.588 inches, about 0.091 inch shorter, and that shorter piece will not reach the 10 inch circle once the corners are turned off. The inside diameter is 9 inches. The inside edge is 2.412 inches. Twelve outer edges in a row, with no kerf, need 32.154 inches of stock.
An 8 inch octagon, ¾ inch wall, cut from one strip with a ⅛ inch kerf. The miter is 22.5°. The outer edge is 3.314 inches. The inside diameter is 6.5 inches and the inside edge is 2.692 inches. Eight edges plus seven kerfs need 27.385 inches of strip.
A 12 inch hexagon, 1 inch wall, no kerf. The miter is 30°. The outer edge is 12 × tan(30°) = 6.928 inches. Six of them need 41.569 inches.
| Ring | Miter | Outer edge | Strip |
|---|---|---|---|
| 10 in, 12 segments, ½ in wall | 15° | 2.679 in | 32.154 in |
| 8 in octagon, ¾ in wall, ⅛ in kerf | 22.5° | 3.314 in | 27.385 in |
| 12 in hexagon, 1 in wall | 30° | 6.928 in | 41.569 in |
What this will not decide
- It is a regular polygon only. An oval stave construction is a different drawing.
- A small angle error closes up on the last segment. Cut a test ring before you commit the good stock.
- The wall is the finished wall. Extra wood for a heavy cut is yours to add, on purpose, before you type the diameter.
Questions
Why is the segment longer than the circumference divided by the count?
The circumference is the curve. The segment you cut is a straight side. For a 10 inch ring of 12 segments, the arc of one twelfth is shorter than the straight edge that reaches the finished circle at the corners. That straight edge is about 2.679 inches. Dividing the circumference by 12 under-sizes the piece you need before the corners are turned away.
Is the miter the same angle as a picture frame?
The setting is the same kind of number: half the change in direction, measured from a square cut. A square frame is 45°. A hexagon is 30°. An octagon is 22.5°. A 12-segment ring is 15°. The frame calculator only does rectangles. The ring calculator does any count from 4 to 64 and also gives the segment length.
Which diameter do I type?
The finished outside diameter, the circle you want after the corners are turned or planed off. If you type a diameter that already includes waste, every segment comes out long. The wall thickness is the finished wall, from that outside face to the finished inside face. Add your own extra if you want wood left for a heavy turning cut.
Does the kerf change the miter?
No. The kerf is the tooth path between segments when they are cut from one strip. It lengthens the strip. It does not change the angle or the length of each finished outer edge.