Woodworking & Construction Calculators

Fine woodworking & cabinetry

Polygon Miter and Segment Length Calculator

For n sides, each miter from square is 180°/n. When the flats of the polygon touch a circle of diameter D, as a turned ring needs, each outside face is D × tan(π/n). When the corners sit on that circle instead, each side is the chord D × sin(π/n).

Inputs

Units
in

Results

Miter from square
22.500°
Outside face
4.971in

Long point to long point; the flats touch the diameter.

Chord
4.592in

Side length when the corners sit on the diameter.

The Result is indicative and educational. It does not constitute a design, a technical opinion or medical advice and does not replace calculations or verification by a person with the appropriate qualifications and authorisations. Before purchase, installation or use in practice – in particular in electrical installations – the Result should be verified by a qualified specialist. Details: Terms of Service, point 4.

Example products

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Calculation method

A regular n-gon is n identical isosceles triangles. The half-angle 180°/n gives the miter (90° − 180°/n from the long edge, which is 180°/n from square) and both lengths: the outside face is twice the flat radius × tan(180°/n), the chord twice the corner radius × sin(180°/n).

Exact formula

Hexagon: miter 30°, face = D / √3, chord = D / 2. For rings, D is the outside diameter of the finished ring. Angle errors add up over the 2n cuts: 0.1° too much on every cut of a 12-segment ring leaves a 2.4° gap at the last joint.

Bowls and gazebos

The same n and D describe a turning ring and an octagonal picnic table. Only the diameter unit changes with the toggle.

Frequently asked questions

Miter 22.5° on an octagon?

Yes. 180/8 = 22.5°. That is the saw setting from a square cut, as read on a miter saw. A miter gauge that reads 90° at square needs 67.5° instead.

Face vs chord?

Use the outside face for segmented and staved rings: the flats then touch the finished outside diameter, so the ring turns round. Use the chord when the corners must land on the circle, such as posts spaced around a gazebo of diameter D.

Does wall thickness matter?

This diameter is outside. For thick staves, some turners size the face at the mid-wall. This tool stays on the outside.

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