Polygon Frame Miter Angle Calculator (Hexagon, Octagon and Other Frames)
The saw angle and the piece lengths for a frame, planter or table top with any number of equal sides, from the width across the flats and the board width.
Results
What this tool does
In a regular polygon of n sides every joint is cut at 180° ÷ n from square: 45° for a square frame, 30° for a hexagon, 22.5° for an octagon. On a miter saw lying flat that is the swing angle; for a box or planter with upright sides it is the tilt of the blade. Each piece has a short edge on the inside of the frame and a long edge on the outside, and the long one is longer by 2 × board width × tan(180° ÷ n) — measure and mark from the long point, as it is the one you cut to. Add a little for the kerf and test the angle on offcuts before cutting the good board.
Formula
saw angle = 180° ÷ n · inside edge = width across flats × tan(180° ÷ n) · long edge = (inside width + 2 × board width) × tan(180° ÷ n)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
n | Number of sides | — |
af | Inside width across the flats (twice the distance from the centre to an inside edge) | mm |
bw | Face width of the board | mm |
MA | Saw angle (miter or bevel) for each joint | ° |
IA | Angle inside each corner | ° |
SI | Length of each piece along its inside (short) edge | mm |
SO | Length of each piece along its outside (long) edge | mm |
OA | Outside width across the flats | mm |
TL | Board needed, long edges added up | mm |
Worked example
- Number of sides6
- Inside width across the flats (twice the distance from the centre to an inside edge)400 mm
- Face width of the board60 mm
- Saw angle (miter or bevel) for each joint30.00 °
- Angle inside each corner120.00 °
- Length of each piece along its inside (short) edge230.9 mm
- Length of each piece along its outside (long) edge300.2 mm
- Outside width across the flats520.0 mm
- Board needed, long edges added up1801 mm
Limitations
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.