Bolt Circle & Sine Bar Calculator
Hole positions on a bolt circle and gauge block stacks for a sine bar — with the odd-count measuring problem and the accuracy that vanishes above 45°.
Work backwards from a measured chord
How accuracy falls away with angle
Same bar, same gauge blocks. Only the angle changes — and the error multiplier is sec(angle), which is why the last column runs away.
Two layouts, one trap each
A sine bar loses accuracy as the angle rises, faster than almost anyone expects. The setting is height = length × sin(angle), so an error in the gauge block stack becomes an angular error of dH ÷ (L × cos angle). That cosine in the denominator is the whole problem: the magnification is sec(angle) — 1.00 at zero, 1.41 at 45°, 2.00 at 60°, and running away to infinity at 90°. On a 5″ bar, a tenth-thou stack error is under 5 arcseconds at 30° and nearly 24 at 80°. Same bar, same blocks, five times the error.
Which is why toolrooms set steep angles from the complement. Above about 45° you stop setting the angle you want and set 90° minus it from the adjacent face. That puts you back on the steep part of the sine curve, where height is sensitive to angle — and the accuracy comes back with it, for no extra equipment.
And an odd bolt circle cannot be measured across. With an even number of holes, opposite holes exist and the distance between their centres is the bolt circle diameter. With an odd number — five-stud wheel patterns above all — no two holes are opposite, so there is nothing to put a rule between. The way out is that the chord between any two holes is D × sin(steps × 180 ÷ count), so one chord and a count of steps gives the diameter exactly.
- Build a gauge stack by clearing the smallest decimal place first. Only one block in the set can do it, and each later step absorbs the whole millimetre it brought along. Working from the largest block down routinely strands a remainder no combination reaches.
- Wring the blocks, don't stack them. A properly wrung joint adds no measurable thickness; a film of oil or a speck of grit adds several microns each — enough to swamp the accuracy you bought the blocks for.
- Measure adjacent holes rather than distant ones where you can. The same absolute measuring error is a smaller proportion of a short distance — but only if you can find the hole centres reliably, which is easier on a drilled plate than on a wheel.
- Bolt circle position tolerance is usually a diametral zone, not plus-or-minus on X and Y. That allows about 40% more error in the diagonal directions than a square tolerance would.
How to use
- For a bolt circle, enter the hole count and diameter, or pick a wheel pattern.
- Use the reverse panel to get the diameter from a measured chord.
- For a sine bar, pick the bar length and the angle you want.
- Above 45 degrees, set the complement instead — the tool works out both.
Frequently asked questions
Why does a sine bar get less accurate at steep angles?
Because the setting is height equals length times sine of the angle, so an error in the gauge block stack becomes an angular error of that height error divided by length times the cosine of the angle. The cosine in the denominator means the magnification is the secant of the angle: one at zero degrees, 1.41 at forty-five, two at sixty, and running away to infinity at ninety. The sine curve has flattened out, so a large change in angle barely changes the height any more.
How much difference does the angle actually make?
On a five inch bar, a tenth of a thousandth of an inch of stack error is under five arcseconds at thirty degrees and nearly twenty-four at eighty. Same bar, same blocks, five times the error — purely because of where you are on the sine curve. That is why the angle you are setting matters as much to the result as the quality of the equipment you are setting it with.
What is the complement trick?
Above about forty-five degrees, toolrooms stop setting the angle they want and set ninety minus it instead, working from the adjacent face of the part. That puts the setup back on the steep part of the sine curve where height is genuinely sensitive to angle, and the accuracy returns with it. It costs nothing but a different reference face, and it can be several times better at steep angles.
Why can I not measure a five-stud bolt pattern across?
Because with an odd number of holes no two are opposite each other, so there is nothing to put a rule between. Even patterns have opposite pairs and the distance between two opposite hole centres is the bolt circle diameter directly. Odd patterns do not, which is why five-stud wheel patterns are the ones people most often get wrong when trying to identify a wheel.
How do I find the diameter of an odd bolt pattern then?
Use the chord. The straight-line distance between any two holes is the diameter times the sine of steps times a hundred and eighty divided by the hole count. So measure between the two most widely separated holes, note how many steps apart they are, and divide by that sine to get the diameter back exactly. On a five-stud pattern the widest pair is two steps apart, and the sine works out at about 0.951.
What is the stud spacing on a 5x114.3 wheel?
About 67.2 millimetres between adjacent stud centres, and about 108.7 between the two furthest apart. Those are figures you can check with a rule, which makes them a useful sanity test when identifying an unfamiliar wheel. A 5x100 pattern gives about 58.8 millimetres between adjacent studs by the same calculation.
How should I build a gauge block stack?
Clear the smallest decimal place first, because only one block in the set can do it. Each of those blocks also carries a whole millimetre, which the later steps absorb. Working the other way round, taking the largest block that fits and repeating, routinely strands a remainder that no combination of the remaining blocks can reach. Use as few blocks as possible, since every joint is another chance to introduce error.
Does it matter how gauge blocks are joined?
A great deal. They should be wrung together, sliding one across the other under light pressure until they adhere, which is held by molecular attraction and adds no measurable thickness. Simply stacking them loose leaves films of air and oil, and a speck of grit adds several microns on its own — easily enough to swamp the accuracy the blocks were bought for in the first place.
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