A bolt circle is a set of holes spaced evenly around a circle. It sounds trivial and mostly it is, but three things make it worth calculating rather than eyeballing: the coordinates have to be exact if a machine is going to drill them, the chord between neighbouring holes is what you actually measure to check the work, and an odd hole count breaks the obvious across-the-part measurement that everyone reaches for first. The bolt circle calculator above handles all three, and will also work backwards from a measured chord to the bolt circle diameter of an existing part.
Arb Digital publishes this as a layout and inspection tool. It is pure geometry: it places holes on a circle and reports the distances between them. It has nothing to say about whether a bolted joint is adequate, how many bolts a flange needs, or what any of them should be tightened to — those are engineering questions decided by the applicable design standard and by whoever is responsible for the assembly.
What This Bolt Circle Calculator Does
It produces a full coordinate list, hole by hole, in Cartesian X and Y relative to a centre you choose. Alongside that it reports the angular step, the straight-line chord between adjacent holes, the arc distance along the circle, and the widest across-the-pattern measurement, which behaves differently depending on whether the hole count is odd or even.
It also reports the ligament, the material remaining between two neighbouring holes, from the chord and the hole diameter you enter. That number is not used in placing the holes, but it is the one that tells you at a glance whether a pattern is physically sensible: a ligament approaching zero means the holes are about to break into one another, and a negative one means they already have.
The chord mode is the reverse-engineering case. Given a part with an unknown bolt circle, measure centre to centre between two neighbours, count the holes, and the diameter follows. That is far more accurate on an odd-hole pattern than trying to measure across the part, because on an odd pattern no hole sits opposite any other.
How to Use It
- Confirm the diameter is centre to centre. A bolt circle never includes the hole size, and mixing the two in is the classic error.
- Set the start angle to match your drawing. Zero degrees puts the first hole at three o'clock; 90 puts it at twelve.
- Offset the centre if the pattern is not at your datum. The coordinates shift with it, so they can be used directly.
- Check the chord against the part. Measuring between adjacent holes is the fastest and most reliable verification of a drilled pattern.
- Look at the ligament before committing. It is the number that tells you whether the pattern can physically exist in the material.
The Formula / How It's Calculated
With a bolt circle diameter D, radius r = D/2, hole count N and start angle α, hole number k sits at an angle of α + 360k / N degrees, and its coordinates are x = Cx + r cos(angle), y = Cy + r sin(angle). That is the standard parametric description of a circle, as set out at Wolfram MathWorld's Circle entry.
The chord between two adjacent holes is c = D × sin(180° / N), and running that backwards gives the diameter from a measurement: D = c / sin(180° / N). The arc distance along the circle between neighbours is πD / N, always slightly longer than the chord, and the gap widens as the hole count falls.
The across-the-pattern measurement splits by parity. With an even hole count, holes sit in diametrically opposite pairs, so the widest centre-to-centre distance is exactly the bolt circle diameter D. With an odd count, nothing is opposite anything, and the widest pair is separated by D × cos(90° / N), which is always slightly less than D. That is why a five-stud wheel or a three-bolt flange cannot be measured across in the obvious way.
Worked example with the loaded values. A 100 diameter bolt circle with 5 holes starting at 0 degrees steps every 72 degrees. The chord is 100 × sin 36° = 58.7785, the arc is π × 100 / 5 = 62.8319, and the widest pair is 100 × cos 18° = 95.1057. The first hole lands at (50, 0), the second at (15.451, 47.553), the third at (−40.451, 29.389). With 10 diameter holes, the ligament between neighbours is 58.7785 − 10 = 48.7785.
The Odd Hole Count Problem
This is where most bolt circle mistakes happen, and it is worth understanding rather than memorising. On an even pattern every hole has an exact partner across the circle, so a caliper across two opposite holes measures the bolt circle directly. On an odd pattern there is no partner. The furthest hole from any given one sits half a step off the diameter, so the measurement you take is short of the true bolt circle by a factor of the cosine of half the angular step.
For five holes that shortfall is about five per cent — 95.1 measured on a 100 circle. For three holes it is far worse, at about 13 per cent. Someone measuring across a three-bolt pattern and writing down the answer as the bolt circle will be out by an amount no machining tolerance will absorb. The reliable method on an odd pattern is to measure the chord between two adjacent holes and let the tool convert.
There is a second wrinkle in the field. Measuring centre to centre between two holes is awkward with a caliper, so the usual technique is to measure between the near edge of one hole and the far edge of the next. That gives the centre distance directly, as long as the two holes are genuinely the same size. If they are not, or if one is worn oval, the measurement inherits the difference. The clearance hole calculator covers the hole sizes themselves.
Coordinates, Machines and Where the Datum Sits
The coordinates this tool prints are Cartesian, relative to the centre you nominate. Most CNC controls will take a bolt-hole cycle directly — you give the centre, the radius, the start angle and the count, and the control generates the points — so the coordinate list is more often used for verification, for manual machines, for marking out, and for drawings where the holes have to be dimensioned individually.
The convention worth stating is the angular one. This page measures the start angle anticlockwise from the positive X axis, which is the standard mathematical convention and what most controls expect. Some drawings measure clockwise from twelve o'clock instead, in the manner of a compass bearing. Those two conventions produce mirror images of one another, which on a symmetrical pattern is invisible and on an asymmetric assembly is a scrapped part. If you work in polar coordinates natively, our polar coordinates converter handles the conversion, and the coordinates converter deals with other systems.
Unit discipline matters here more than the arithmetic does. The calculation is unit-agnostic, but a coordinate list is not self-describing, and a table of numbers that could be either millimetres or inches is a genuine hazard on a shop floor. Label them. The conventions for doing so are set out in NIST Special Publication 811, the Guide for the Use of the International System of Units.
Bolt Circles Are Not Just Flanges
The same geometry turns up in places that do not look related. Wheel studs on a vehicle are quoted as a count and a bolt circle, and the odd-count problem above is exactly why five-stud patterns are so often recorded wrongly. Pipe flanges use bolt circles set by the flange standard, with the count and the drilling fixed so that flanges from different makers mate. Rotary tables, index plates, lantern rings, sprocket hubs, clock faces and the hole pattern in a speaker baffle are all the same calculation.
Two design points come with it. The first is straddling the centreline: on a flange that has to be oriented, holes are conventionally arranged so that a pair straddles the vertical rather than one sitting on it, which is a start-angle decision of half a step. The second is that an even count lets a component be fitted in more than one rotational position, which is sometimes wanted and sometimes precisely what you are trying to prevent — a deliberately unequal pattern is the usual way of stopping a part being assembled the wrong way round.
None of that is a strength calculation. How many bolts a joint needs, what size they should be, and what preload they carry are questions for the applicable design standard and for a qualified engineer. Our bolt torque calculator covers preload and tightening torque with those limits stated, and the polygon calculator covers the regular polygon whose vertices your holes sit on.
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Browse Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Measuring across an odd hole pattern — no hole is opposite any other, so the reading is short of the true bolt circle by the cosine of half a step, which is 13 per cent on three holes.
- Quoting a bolt circle edge to edge — it is always centre to centre, and including the hole diameter shifts every coordinate.
- Mixing angular conventions — anticlockwise from three o'clock and clockwise from twelve produce mirror images, which is invisible on a symmetrical pattern and fatal on an asymmetric one.
- Ignoring the ligament — with a large hole diameter on a small circle the holes break into one another, and the coordinates will not tell you that on their own.
- Leaving the coordinate list unlabelled — a column of numbers that could be inches or millimetres is a real hazard once it leaves your screen.
Related Free Tools From Arb Digital
Size the holes with the clearance hole calculator, tighten the fasteners with the bolt torque calculator, and work the underlying geometry with the circle calculator, the chord calculator and the polygon calculator. For sheet parts, the bend allowance calculator handles flat patterns and the angle cut calculator handles angled cuts. Everything we publish is listed on the free online tools hub.
Frequently Asked Questions
Always through the hole centres. The bolt circle is the circle the centres lie on, so the hole diameter never enters into it. Quoting an edge-to-edge measurement as the bolt circle shifts every coordinate in the pattern.
Because with an odd hole count no hole sits opposite any other. The widest pair is separated by the diameter times the cosine of half the angular step, which is about five per cent short on five holes and thirteen per cent short on three.
Count the holes, measure centre to centre between two adjacent ones, and divide that chord by the sine of 180 degrees over the hole count. The chord mode on this page does it directly, and it is far more accurate than measuring across.
The chord is the straight line between two hole centres, which is what you measure. The arc is the curved distance along the bolt circle itself. The arc is always the longer of the two, and the difference grows as the hole count falls.
Anticlockwise from the positive X axis, so zero is three o'clock and 90 is twelve o'clock. Drawings that measure clockwise from twelve produce a mirrored pattern, which matters on any asymmetric assembly.
It is the material left between two neighbouring holes, the chord minus the hole diameter. As it approaches zero the holes are about to break into one another, and a negative value means the pattern cannot be drilled as specified.
No. It places holes on a circle. Bolt count, size and preload are set by the applicable design standard and by the engineer responsible for the joint.
Yes. Enter the centre X and Y and every coordinate shifts with it, so the list can be used directly against your part datum without further arithmetic.
This tool computes the geometry of a hole pattern from dimensions you supply, for layout and inspection only. It is not a design of any bolted joint, and bolt count, size and preload are matters for the applicable design standard and a qualified engineer.
