Hexagon and octagon calculator: side, flats and corners
Enter the number of sides and the one dimension you have — the side, across the flats or across the corners. This works out the other two, the area, the miter and each frame piece's long- and short-point length.
8 sides is a octagon
Flat side to the flat opposite it
Advanced settings3" boards
Face width of the frame stock, for the short point of each piece
A octagon, 24" across the flats
Across corners
26"
Side length
9 15/16"
Miter, each end
22.5°
Corner
135.0°
Area
3.31 sq ft
Cutting 8 frame pieces from 3" boards
- Long point to long pointThe outside edge of each piece — the side length
- 9 15/16"
- Short point to short pointThe inside edge, where the miters meet the opening
- 7 7/16"
- MiterBoth ends of every piece, the cuts mirror images
- 22.5° off square
- Opening left inside19 1/2" across its corners
- 18" flats
- Stock, end to endAll the pieces at their long points, before kerf — flip the board between cuts to nest them and save most of a piece
- 79 1/2"
All 2 checks passShow the numbers
- Miter inside the 50° most saws reach — passes22.5° on the miter scale, both ends of every piece.
- The frame leaves an opening — passes12" from centre to flat, less a 3" board, leaves an opening 18" across the flats.
- Miter inside the 50° most saws reach — passes22.5° on the miter scale, both ends of every piece.
- The frame leaves an opening — passes12" from centre to flat, less a 3" board, leaves an opening 18" across the flats.
Miter check, at true size
Print at 100%, check the ruler, then set a bevel gauge or a test cut against the two lines before you cut 8 pieces. Every degree off on one cut is 16° of gap when the octagon closes.
Put a tape on the printed ruler. If it does not read exactly 6 inches, the print was scaled — set the print dialog to 100% or “Actual size” and print it again. Do not cut wood from a template that failed this.
Baseline registers against your saw's fence or a straightedge; the second line is the miter cut.
Printing this properly
In the print dialog, set the scale to 100% or “Actual size” and turn off “Fit to page”. Measure the printed ruler with a tape before you trust the angle.
Worked example: an octagon 24" across the flats
An octagon frame 24" across the flats is 26" across the corners, with sides of 9 15/16" and 3.31 sq ft inside its outline.
Cut from 3" boards, each of the 8 pieces is 9 15/16" long point to long point and 7 7/16" short point to short point, mitered 22.5° both ends, leaving an opening 18" across.
The same 24" across the flats as a hexagon is 27 11/16" across the corners with 13 7/8" sides and a 30.0° miter.
- Across flats
- 24"
- Across corners
- 26"
- Side
- 9 15/16"
- Miter
- 22.5°
- Short point
- 7 7/16"
These are the figures the calculator above opens on. Change any field and it works the same problem for your numbers.
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About polygon sizes
- How do I work out the side length of an octagon from its width?
- Multiply the width across the flats by tan(180° ÷ 8), which is 0.4142. An octagon 24" across the flats has sides of 9 15/16". For any regular polygon it is the same rule with its own number of sides: the side is the across-flats width times tan(180° ÷ N), or the across-corners width times sin(180° ÷ N).
- What is the difference between across the flats and across the corners?
- Across the flats is flat side to the flat side opposite it — the width of the circle that fits inside. Across the corners is point to opposite point — the circle the polygon fits inside. They differ by 1 ÷ cos(180° ÷ N): a hexagon 24" across the flats is 27 11/16" across the corners. Know which one a plan means before cutting: at that size the two differ by 3 11/16". On a polygon with an odd number of sides there is a point opposite every flat, so the width a tape reads across it is half the one plus half the other.
- What miter angle do I cut for a hexagon or an octagon?
- 180° divided by the number of sides, on both ends of every piece: 30.0° for a hexagon and 22.5° for an octagon. It is the same flat-trim miter the miter angle calculator gives for a corner of 135°, an octagon's inside angle. And the errors add up: an octagon frame is 16 cuts, so a quarter of a degree out on each, all the same way, closes the frame 4° short.
- What are the long point and the short point of a frame piece?
- A mitered piece is a trapezoid: the long point to long point length is its outside edge, which is the polygon's side, and the short point to short point length is its inside edge. The difference is twice the board width times tan(180° ÷ N). For an octagon 24" across the flats made of 3" boards, each piece is 9 15/16" long point to long point and 7 7/16" short point to short point, and it leaves an opening 18" across the flats.
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