Hex and square: across flats and across corners
From the across-flats size get the across-corners size and the round bar you need. From a bar diameter get the hex, square or octagon it will make.
Quick answer
Across corners E = S / cos(180°/n): hex E = 1.1547 × S, square E = 1.4142 × S. Example: 3/4″ hex → 0.866″ across corners; 17 mm hex → 19.63 mm, so you need at least a Ø20 mm bar.
Result
- Across corners (circumscribed circle)
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- Minimum round bar
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- nearest standard size
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- Full round from this bar without flat spots — up to
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- Flat width
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Leave some stock for finishing the flats: a bar exactly the across-corners size leaves sharp corners with nothing to clean up.
Who needs it and why
A 3/4″ hex measures about 0.866″ across the corners. Use a 13/16″ bar and the corners stay round and unfinished; use a 1-1/4″ bar and you waste metal and time. The calculator suggests the nearest bar that works.
- Choosing stock — which round bar to order for a hex or square on the part.
- Parts from hex bar — fittings, nuts, adapters: the largest full round you can turn without flat spots.
- Using what's on the shelf — which wrench size a bar you have will give.
- Chamfering a hex — nut corners are usually chamfered at a diameter just under the across-flats size; across-corners shows how much comes off.
The formulas
- across corners
E = S / cos(180° / n): hex 1.1547·S, square 1.4142·S, octagon 1.0824·S - from a bar Ø D
S = D · cos(180° / n) - flat width
S · tg(180° / n)
The round part of the workpiece, chamfers and faces next to the hex come straight out of Turn G-code — turning, facing and Chamfer/Radius cycles with a 2D simulator.