Bolt Circle Calculator (Hole Coordinates)
X and Y coordinates for any bolt circle, plus the chord between holes - and the reverse, working out an unknown bolt circle from a measurement.
How to use this calculator
- 1For laying out, enter the bolt circle and hole count and take the coordinates straight into the programme.
- 2For identifying an unknown flange or wheel, switch to the reverse mode and measure centre to centre between two adjacent holes.
- 3To measure centre to centre with a caliper, go from the far edge of one hole to the near edge of the other - the radii cancel.
- 4On an even hole count, measure straight across opposite holes instead; it is one measurement and no trig factor to amplify the error.
- 5Check the first two coordinates against the chord before cutting. It catches almost every possible mistake.
How the calculation works
Angle between holes = 360 / N
X = centre X + (D/2) x cos(start + n x 360/N)
Y = centre Y + (D/2) x sin(start + n x 360/N)
Chord between adjacent holes = D x sin(180/N)
So D = chord / sin(180/N)- D
- Bolt circle diameter, through the hole centres. Also called the pitch circle diameter, or PCD
- N
- Number of holes, evenly spaced
- Chord
- The straight line distance between two adjacent hole centres. It is what you can actually measure on an odd pattern
- Sagitta
- How far the arc bulges from the chord. Useful when laying out from a straight edge
Angles run anticlockwise from the positive X axis, which is the standard convention in CNC and in mathematics alike. Zero degrees is three o'clock.
On an even hole count, the distance across two opposite holes is exactly the bolt circle diameter, because they are a diameter apart by definition.
For five holes the factor 1/sin(36°) = 1.7013 converts an adjacent-hole measurement to the bolt circle. That is the number on every wheel PCD chart.
Worked example
Six holes on a four inch circle
- 1.Six holes means 60 degrees apart, on a radius of 2.0000.
- 2.The first is at 0 degrees: X = 2.0000, Y = 0.0000. The second at 60: X = 1.0000, Y = 1.7321.
- 3.The chord is 4 x sin(30°) = 4 x 0.5 = 2.0000 exactly - a hexagon's side equals its circumradius, which is the neat case.
- 4.And because six is even, measuring across opposite holes gives 4.0000, the bolt circle itself.
Result: 60° apart, 2.0000 between adjacent holes
Identifying a five hole wheel pattern
- 1.Five holes, so there is nothing opposite anything and a caliper across the middle is useless.
- 2.Measuring centre to centre between two adjacent studs gives 2.6417 in.
- 3.D = 2.6417 / sin(36°) = 2.6417 / 0.58779 = 4.4943 in.
- 4.That is 114.3 mm - the 5x114.3 pattern, one of the commonest there is. The 1.7013 factor on the chart is just 1/sin(36°).
Result: 4.4943 in - the 5 x 114.3 mm pattern
A three hole pattern, rotated
- 1.Three holes, 120 degrees apart, starting at twelve o'clock rather than three.
- 2.Hole 1 at 90°: X = 0.0000, Y = 3.0000. Hole 2 at 210°: X = -2.5981, Y = -1.5000. Hole 3 at 330°: X = 2.5981, Y = -1.5000.
- 3.The chord is 6 x sin(60°) = 5.1962 - much larger than the radius, because with only three holes the arc between them is a third of the circle.
- 4.To find this pattern by measurement you would multiply a 5.1962 chord by 1/sin(60°) = 1.1547.
Result: 120° apart, 5.1962 between holes
An eight hole flange measured across
- 1.Eight holes is even, so holes sit directly opposite each other.
- 2.A measurement straight across opposite hole centres is the bolt circle - 7.5000 in, no trig needed.
- 3.The chord between adjacent holes would be 7.5 x sin(22.5°) = 2.8701.
- 4.Using the across-the-middle measurement is more accurate here: the chord method would multiply any measuring error by 2.61.
Result: 7.5000 in directly - the even-count shortcut
Laying one out is the easy half
Putting N holes evenly on a circle is one line of trigonometry. Divide 360 by the hole count for the angular step, then each hole sits at the radius times the cosine and sine of its angle, offset by wherever the centre is.
The conventions are worth stating because they are where the mistakes happen. Angles run anticlockwise from the positive X axis, so zero degrees is three o'clock and ninety is twelve. Bolt circle diameter is through the hole centres, not to the outside of the flange or the edge of the holes. And a great many patterns are specified by a diameter but drawn from a radius, which is the single commonest way to end up with a part exactly twice the size it should be.
A check that costs nothing: after generating the coordinates, confirm the distance between the first two holes matches the chord formula. That one comparison catches a wrong hole count, a radius-diameter swap, and a degrees-versus-radians error, which between them account for nearly every bolt circle that comes out wrong.
The problem is measuring one you did not make
The situation people actually face is the reverse: an unfamiliar flange, an old casting, a wheel with no markings, and the need to know what bolt circle it is.
With an even number of holes it is easy, because every hole has one directly opposite. Measure centre to centre across the middle and that distance *is* the bolt circle - a pair of opposite points on a circle are a diameter apart by definition. One measurement, no arithmetic.
With an odd number, there is nothing opposite anything. A caliper laid across a five hole pattern lands on a hole on one side and solid metal on the other. This is exactly why odd counts have a reputation for being awkward, and why so many people end up estimating.
The way through is the chord. Two adjacent holes subtend an angle of 360/N at the centre, so the straight distance between them is D times the sine of half that: D x sin(180/N). Measure the chord and divide. For five holes the divisor is sin(36°) = 0.58779, so the multiplier is 1.7013 - the number printed on every wheel bolt pattern chart, invariably without the explanation.
Getting a centre to centre measurement with a caliper
Hole centres are not physical features, so they cannot be touched directly. But there is a clean trick when both holes are the same size, and it deserves to be better known.
Measure from the *far* edge of one hole to the *near* edge of the other. Going out to the far edge adds one radius; stopping at the near edge subtracts one. They cancel exactly, and the reading is the centre distance with no correction and no assumption about hole size.
The alternatives both need a correction. Outside edge to outside edge gives centre distance plus one hole diameter. Inside edge to inside edge gives centre distance minus one hole diameter. Both work, but both require knowing the hole size precisely, and on a worn or chamfered hole that is its own source of error.
One caution about the chord method generally: it amplifies measurement error. Dividing by sin(36°) multiplies whatever you measured - and whatever you got wrong - by 1.7. On a twelve hole pattern the factor is 3.86. Where an across-the-middle measurement is available, it is the better one.
What this assumes, and where it stops
Assumptions
- Holes are evenly spaced around a true circle.
- The bolt circle diameter passes through hole centres.
- Angles are measured anticlockwise from the positive X axis, the standard CNC and mathematical convention.
- Measurements entered for the reverse calculation are centre to centre.
Limitations
- It does not handle unevenly spaced patterns, or patterns with one hole deliberately offset for orientation.
- It does not check that the holes fit - flange thickness, hole size and edge distance are separate questions.
- The chord method amplifies measurement error by the inverse of the sine factor, which is significant on high hole counts.
- It says nothing about tolerance, positional accuracy or true position callouts.
- For very high hole counts, cumulative indexing error on a rotary table matters more than the coordinates do.
Common questions
How do I calculate bolt circle coordinates?
Divide 360 by the number of holes for the angular step. Each hole is at X = centre X + radius x cos(angle) and Y = centre Y + radius x sin(angle), with the angle increasing by the step each time. Angles run anticlockwise from the positive X axis, so zero degrees is three o'clock.
How do I find the bolt circle of a 5 hole pattern?
Measure centre to centre between two adjacent holes and multiply by 1.7013, which is 1/sin(36°). A five hole pattern has nothing opposite anything, so there is no measurement straight across to take. A 2.6417 in chord gives 4.4943 in, which is the common 5 x 114.3 mm pattern.
How do I measure centre to centre between two holes?
If both holes are the same size, measure from the far edge of one to the near edge of the other - the radii cancel exactly and the reading is the centre distance. Outside to outside gives centre distance plus one hole diameter; inside to inside gives it minus one.
What is the chord between bolt holes?
The straight line distance between two adjacent hole centres, equal to the bolt circle diameter times sin(180/N). For six holes on a 4 in circle it is 4 x sin(30°) = 2.000 - a hexagon's side equals its circumradius, which is why six hole patterns lay out so neatly.
Why is an even number of holes easier to measure?
Because every hole has one directly opposite it, and two opposite points on a circle are a diameter apart. So one measurement straight across gives the bolt circle with no calculation. It is also more accurate than the chord method, which multiplies any measuring error by the inverse of the sine factor.
Sources
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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