Turning a bolt circle into coordinates
A bolt circle is described in polar terms — a radius and a set of angles — while a milling machine works in rectangular X and Y. Converting between them is the whole job, and it is two lines of trigonometry: X is the radius times the cosine of the angle, Y is the radius times the sine, both measured from the circle's centre.
The angles themselves come from dividing the circle evenly. With n holes the step is 360 ÷ n, so six holes sit 60 degrees apart and eight sit 45 apart. Each hole's angle is the starting angle plus however many steps it is around the circle.
The convention used here is the standard mathematical one: angles measured from the positive X axis, increasing counter-clockwise. Zero degrees puts the first hole at three o'clock. If your drawing measures from twelve o'clock instead, enter 90 as the start angle.
Diameter or radius, and the error that follows
Drawings almost always specify a bolt circle by its diameter — a 4-inch bolt circle, or a 100 mm PCD. The trigonometry needs the radius. Feeding the diameter straight into the formula produces a pattern exactly twice the size, correctly spaced and completely wrong.
It is worth checking which you have before cutting, because the error is not obvious from the coordinate list — every hole is in the right place relative to the others. The quickest confirmation is the first hole at zero degrees: its X coordinate should be the radius, so on a 4-inch bolt circle it should read 2.0000. If it reads 4.0000, the diameter went in where the radius was wanted.
Worked example: six holes on a 4 inch bolt circle
Six evenly spaced holes, 4-inch bolt circle diameter, first hole at three o'clock, pattern running counter-clockwise, centre at the origin.
- Radius. 4 ÷ 2 = 2.0000 inch.
- Angular step. 360 ÷ 6 = 60 degrees.
- Hole 1 at 0°. X = 2 × cos 0 = 2.0000, Y = 2 × sin 0 = 0.0000.
- Hole 2 at 60°. X = 2 × 0.5 = 1.0000, Y = 2 × 0.86603 = 1.7321.
- Hole 3 at 120°. X = −1.0000, Y = 1.7321.
- Hole 4 at 180°. X = −2.0000, Y = 0.0000.
- Holes 5 and 6 mirror holes 3 and 2 below the axis.
The chord check: 2 × 2 × sin(180 ÷ 6) = 4 × sin 30° = 2.0000 inch between adjacent centres. For six holes specifically the chord equals the radius, because six equilateral triangles fit exactly around a circle — a useful sanity check that costs nothing to remember.
Checking the pattern after you drill it
Measuring a bolt circle directly is awkward, because the centre is usually not a feature you can put a caliper on. The chord between adjacent holes is far easier: measure between two neighbouring hole centres, and compare against 2R sin(180°/n).
For an even hole count there is a second check that is easier still. Holes directly opposite each other are exactly one bolt circle diameter apart, so measuring across the pattern from hole 1 to hole 4 on a six-hole circle should give the bolt circle diameter itself. Odd counts have no opposite hole, which is why the chord is the general method.
Both checks measure centre to centre. With holes of equal size you can measure edge to opposite edge for the same result, or measure inside-to-inside and add one hole diameter — which is how a pattern usually gets checked with nothing but a caliper.
Chord between adjacent holes, per inch of bolt circle diameter
| Holes | Angle apart | Chord factor | On a 4 in circle |
|---|---|---|---|
| 3 | 120° | 0.86603 | 3.4641 |
| 4 | 90° | 0.70711 | 2.8284 |
| 5 | 72° | 0.58779 | 2.3511 |
| 6 | 60° | 0.50000 | 2.0000 |
| 8 | 45° | 0.38268 | 1.5307 |
| 10 | 36° | 0.30902 | 1.2361 |
| 12 | 30° | 0.25882 | 1.0353 |
The chord factor is sin(180°/n). At six holes it is exactly 0.5, so the chord equals the bolt circle radius.
Zero degrees is three o'clock, not twelve
This calculator uses the standard convention: angles measured from the positive X axis, increasing counter-clockwise. A start angle of 0° puts the first hole at three o'clock. Many drawings put the first hole at top dead centre instead — enter 90° as the start angle for that, and the whole pattern rotates with it.
Laying out the pattern
- Confirm diameter against radius. The commonest error, and the resulting pattern looks internally consistent so it is easy to miss.
- Pick the start angle from the drawing, not from habit. A pattern rotated 30 degrees is as wrong as one sized incorrectly, and often harder to spot.
- Check the chord before drilling all of them. Spot the first two, measure between them, and confirm against the chord figure.
- Watch the web between holes. On a small circle with large holes, adjacent holes can break into each other. The chord minus one hole diameter is the material left.
- Match the direction to the drawing. Clockwise and counter-clockwise numbering give mirror-image patterns wherever the holes are not symmetrical.
Related setup work
Most controls have a bolt-circle canned cycle, and where one exists it is the right tool — this page is for manual machines, for programming from scratch, and for checking what a control produced. On a rotary table the coordinates matter less than the angular step, which is the primary output here.
Once the holes are located, the rest of the job is drilling and tapping them. The tap drill calculator gives the pre-drill for a target thread engagement, the tapping speed and feed calculator covers the tapping cycle, and the bolt torque calculator handles what the finished fasteners get tightened to. For angular setups that are not circular patterns, the sine bar calculator is the companion tool.
Terms used here
- Bolt circle diameter
- The diameter of the imaginary circle the hole centres lie on. Called PCD, pitch circle diameter, on metric drawings.
- Chord
- The straight-line distance between two adjacent hole centres, 2R sin(180°/n). The practical way to check a drilled pattern.
- Start angle
- Where the first hole sits, measured from the positive X axis counter-clockwise. Zero is three o'clock.
- Web
- The material left between two adjacent holes — the chord minus one hole diameter. Goes to zero when holes break into each other.