BIZOMAT MAKEOS · Geometry
Regular Polygon and Hexagon Sizes
What are the across-flats, across-corners, side and area of this regular polygon?
Convert between side length, across-flats and across-corners sizes of a regular polygon such as a hexagon or square, with area and angles.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Number of sides | sides | 6 for a hexagon, 4 for a square, 8 for an octagon. |
| Known dimension | choice | Which dimension the drawing or standard specifies. |
| Size of the known dimension | mm | The specified dimension in millimetres. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Number of sides
- What it is
- How many equal sides the regular shape has, for example a hexagon bar, square drive or octagon stock.
- Units and format
- Enter a number in sides: sides. The form accepts 3 sides to 1000 sides. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 6 sides as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The tool assumes all sides and angles are equal; a stretched or irregular shape will not match.
What this means: Known dimension
- What it is
- Which size you already know: across flats (between two opposite sides), across corners (between two opposite corners) or the length of one side.
- Units and format
- Choose one of 3 options from the list: Across flats (inscribed circle diameter); Across corners (circumscribed circle diameter); Side length.
- Example
- The form opens with “Across flats (inscribed circle diameter)” selected as a worked example. Choose the option that matches your case.
- Get it right
- Spanner sizes and hexagon bar are normally specified across flats; across corners is the round stock needed to make the shape.
What this means: Size of the known dimension
- What it is
- The measured or specified size of the dimension chosen above: across flats, across corners or one side length.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 24 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- All lengths scale in proportion to it and the area with its square.
- Get it right
- Check it matches the dimension type selected; an across-corners value entered as across flats gives every result too large.
Method and formulas
- Across flats F = s ÷ tan(π/n); across corners C = s ÷ sin(π/n), where s is the side and n the number of sides.
- Area = n × s² ÷ (4 tan(π/n)); interior angle = 180° × (n − 2) ÷ n.
- Example: a hexagon 24 mm across flats is 27.713 mm across corners with 13.856 mm sides and an area of 498.83 mm².
Limitations
- Sharp corners are assumed; chamfered or rounded corners reduce the across-corners size and area.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the bar size that will yield a hexagon or square across flats
- Check spanner flats and corner clearance
- Lay out polygon pockets
Who it is for
- Design engineers
- Toolmakers
- Estimators
What you get
- Across flats
- Across corners
- Side length
- Area
- Interior angle
- Perimeter
Interpreting the result
- Round bar stock must be at least the across-corners size to machine a full polygon.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 76, 81–82. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Polygons, ellipses and inclined views
In Drafting Geometry by the Numbers
What it covers: Draw regular polygons by across-flats or across-corners size, construct the ellipses that circles become on inclined faces and oblique cuts, and estimate an ellipse's perimeter.
Method explained here · BIZOMAT original book
Polygons on the bench
What it covers: Mark hexagons, octagons and pentagons by construction, see why stepping sides round a polygon fails at the turns, and check a finished polygon by its sides, diagonals and flats.
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