Layout and shop math
Hex, square and octagon across corners calculator
Hex across corners = across flats x 1.1547; a 3/4 in hex measures 0.8660 in across corners.
Technical review pending. Formulas and values are cited. This notice comes down after a machinist review.
Across corners
in
AC = AF x 1.1547 = 0.75 x 1.1547 = 0.8660 in
0.8660 in (22.00 mm). Sharp-corner value.
Minimum round stock
in
D min = AC = 0.8660 in (sharp corners)
Round stock at least this size, plus stock for the finish pass (rule of thumb).
Source: Machinery's Handbook (distance across corners of squares and hexagons); plane geometry
How it works
Hex: AC = AF / cos(30°) = 1.1547 x AF
| Symbol | Meaning | Unit |
|---|---|---|
| AF | Across flats (wrench size) | in or mm |
| AC | Across corners, sharp corners | in or mm |
| S | Side of a square | in or mm |
| a, b | Sides of a rectangle | in or mm |
| D | Round stock diameter | in or mm |
| Shape | Formula | Factor |
|---|---|---|
| Hex across corners | AC = AF / cos(30°) | 1.1547 |
| Square across corners | AC = S x sqrt(2) | 1.4142 |
| Octagon across corners | AC = AF / cos(22.5°) | 1.0824 |
| Largest hex from round | AF = D x cos(30°) | 0.8660 |
| Largest square from round | S = D / sqrt(2) | 0.7071 |
Factors are unitless: multiply by the size in inches or millimeters. Rectangle diagonal = sqrt(a^2 + b^2).
Worked example: 3/4 in hex
A 3/4 in hex, 0.750 in across flats.
- cos(30°) = 0.86603, so 1 / 0.86603 = 1.1547.
- AC = 0.750 x 1.1547 = 0.8660 in.
Result: 0.8660 in (22.00 mm) across corners.
Worked example: 13 mm hex
A 13 mm hex.
- AC = 13 x 1.1547 = 15.01 mm.
Result: 15.01 mm (0.5910 in) across corners.
Worked example: square, octagon and rectangle
A 1.000 in square, a 1.000 in octagon and a 3 x 4 in rectangle.
- Square: 1.000 x 1.4142 = 1.4142 in.
- Octagon: 1.000 x 1.0824 = 1.0824 in.
- Rectangle: sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.0000 in.
Result: 1.4142 in, 1.0824 in and 5.0000 in corner to corner.
Worked example: largest hex from round bar
Mill a hex from 1.000 in round.
- AF = 1.000 x 0.8660 = 0.8660 in.
- The largest square is 1.000 x 0.7071 = 0.7071 in on a side.
Result: 0.8660 in hex or 0.7071 in square, with sharp corners touching the round.
Shop notes
- Commercial hex bar has broken corners, so a measured AC reads under the sharp-corner value.
- Round stock to mill a hex or square: at least AC, plus stock for the finish pass (rule of thumb).
- On a dividing head, hex flats are 60° apart: 6 2/3 turns of the crank on a 40:1 head.
FAQ
What is the distance across the corners of a 3/4 in hex?
0.866 in: 0.750 x 1.1547 = 0.8660 in, for sharp corners.
What round stock does a 1/2 in hex need?
At least 0.5774 in (0.500 x 1.1547), plus stock for the finish pass.
What is 13 mm hex across corners?
15.01 mm: 13 x 1.1547.
What is the biggest square you can mill from 1 in round?
0.7071 in on a side (1.000 x 0.7071). The biggest hex is 0.8660 in across flats.
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