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Tool Belt Calc

Cuts

Polygon and Segment Calculator

Miter angle and segment length for any regular polygon.

Your measurementsEvery assumption editable01

The inside diameter, measured flat face to opposite flat face.

SegmentsCalculated live02

Result

22.5° saw setting

8 sides — 9.9411 in each, mitred 67.5°

Sides
8
Interior angle
135°
Mitre per endhalf the interior angle
67.5°
Saw scale setting180 ÷ 8, measured off square
22.5°
Side lengthlong point to long point
9.9411 in
Across the flats
24 in
Across the corners
25.9774 in
Perimeter
79.529 in
Area3.314 sq ft
477.17 sq in

The saw setting is simply 180 divided by the number of sides — 45 for a square, 30 for a hexagon, 22.5 for an octagon. That shortcut is worth memorising, and the full interior angle is given above so you can check which convention your saw uses.

Side length is measured long point to long point, which is the dimension you cut to. Measuring to the short points instead leaves every segment undersized by twice the mitre offset.

Cumulative error is the real enemy on a segmented ring. A tenth of a degree per joint is invisible on one cut and closes badly by the twelfth, so dial the setting in on a test ring before cutting the real stock.

Octagon columns, hexagonal planters, segmented rings and staved barrels all reduce to the same arithmetic. Enter the number of sides and the across-flats dimension and this gives the mitre and the segment length.

Why use this tool?What it does differently03

Why use this tool?

The 180-over-n shortcut

Saw setting is simply 180 divided by the number of sides. Worth memorising, and shown alongside the full interior angle.

Long point to long point

Side length given as the dimension you actually cut to, not the short points.

Both diameters

Across the flats and across the corners, since stock and openings get specified either way.

Cumulative error flagged

A tenth of a degree per joint is invisible on one cut and closes badly by the twelfth.

How this worksThe method04

How this polygon and segment calculator works

The interior angle of a regular polygon is (n − 2) × 180 ÷ n. Two segments meet at each corner so each is mitred at half that, and the saw — reading off square — shows 180 ÷ n.

Side length comes from the inradius and the central half-angle: twice the inradius times the tangent of 180 ÷ n. The circumradius, and therefore the across-corners dimension, follows from dividing the inradius by the cosine of the same angle.

Cumulative error is the real difficulty with segmented work. Each joint contributes its own small error and they all add up around the ring, so a setting that looks fine on a single test joint can leave a visible gap by the last one.

How to use itStep by step05

How to use it

  1. Step 1: Choose the number of sides

    Eight for an octagon, six for a hexagon, and so on.

  2. Step 2: Enter across the flats

    The inside diameter, flat face to opposite flat face.

  3. Step 3: Set the saw

    To 180 divided by the number of sides — 22.5 for an octagon.

  4. Step 4: Test the full ring

    Dry-assemble all segments before glue. Cumulative error only shows when they are all together.

Example usageWorked figures06

Example usage

An octagon column
Eight sides at 24 in across the flats needs segments 9.9411 in long, mitred 67.5° each end, saw set to 22.5. Across the corners is 25.9774 in.
A hexagonal planter
Six sides gives a 30° saw setting and a 120° interior angle — the easiest polygon to cut accurately because the angle is forgiving.
A twelve-segment ring
Twelve sides gives a 15° saw setting. Twelve joints means twelve chances to accumulate error, which is where careful dry assembly earns its time.
Frequently asked questionsCommon questions07

Frequently asked questions

What is the miter angle for an octagon?

22.5 degrees on the saw scale, which is a 67.5 degree cut measured from the face. The shortcut is 180 divided by the number of sides.

What angle do I cut for a hexagon?

30 degrees on the saw — 180 divided by six. The interior angle is 120 degrees and each segment is mitred at 60.

How do I work out segment length?

Twice the inradius times the tangent of 180 ÷ n, where the inradius is half the across-flats dimension. That gives the long-point-to-long-point measurement.

Why does my ring not close up?

Cumulative error. Each joint carries a fraction of a degree and they add around the ring. Dial the setting in on a full dry assembly rather than a single test joint.

Is across the flats the same as diameter?

It is the inscribed diameter — flat to opposite flat. Across the corners is larger, and both are given because suppliers and drawings use either.

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