Geometry and layout calculator
Bolt Circle Calculator
Lay out equally spaced holes around a pitch circle, choose clockwise or counterclockwise numbering, add a coordinate origin, and get every angle and center position in one table. Circles are wonderfully orderly; drills are where the personality begins.
Sage’s guided lesson
The holes are not arranged around the edge; they are arranged around their centers
A bolt circle is an imaginary circle through the center of every hole. That sounds obvious after someone says it, which is how geometry likes to behave. The calculator turns that circle into angles, chords, and coordinates that you can mark, machine, inspect, or compare with a drawing.
Start with the invisible circle
BCD and PCD describe the same centerline geometry
Bolt circle diameter, often shortened to BCD, is also commonly called pitch circle diameter, or PCD. It is the diameter of the circle passing through the hole centers. It is not the flange diameter, wheel diameter, hub diameter, or the distance between neighboring holes.
The radius used by the coordinate formulas is half the BCD. Every hole center lies exactly that radius from the pattern center in an ideal layout.
This geometry appears on flanges, wheels, sprockets, pulleys, faceplates, bearing retainers, pipe fittings, speaker frames, rotary fixtures, control knobs, and decorative patterns. The fasteners may be different; the circle has seen all of this before.
Sketch the part center and one hole center before measuring anything. Label the center to hole distance as the radius and the full centerline circle as the BCD.
One division and two trigonometric functions
How to calculate every bolt circle coordinate yourself
For n equally spaced holes, the angular step is:
Angular step = 360° ÷ number of holes
X = center X + radius × cos(angle)
Y = center Y + radius × sin(angle)
Clockwise numbering reverses the sign of the angular movement. The starting angle rotates the entire pattern without changing any spacing.
Worked example, five holes on a 4.5 inch BCD
Radius = 4.5 ÷ 2 = 2.25 in
Angular step = 360 ÷ 5 = 72°
Hole 2 at 72°: X = 2.25 × cos(72°) = 0.6953 in
Hole 2 at 72°: Y = 2.25 × sin(72°) = 2.1399 in
Keep more coordinate precision than you expect to mark or machine. Round only at the final stage, because early rounding can shift the pattern away from perfect symmetry.
I know this is off topic, but this reminds me of a joke I heard when teaching this to one of my interns, Why did the stadium get hot after the game? All the fans left. Yeah, I don’t see the connection either, but I think it is cute!
The straight shortcut between two holes
Chord spacing turns the circular pattern into a measurable line
The chord is the straight distance between neighboring hole centers. It is often easier to check with calipers than an angle is to inspect directly.
Adjacent chord = 2 × radius × sin(180° ÷ hole count)
For six equally spaced holes, the angular step is 60 degrees. The adjacent chord is exactly equal to the radius, so a six hole pattern creates a regular hexagon inside the circle.
For four holes, the neighboring centers form a square, and the adjacent chord is the BCD divided by √2. For three holes, the centers form an equilateral triangle. Bolt circles are regular polygons wearing round disguises.
For a complete evenly spaced pattern measured from its true center, the X coordinates add to zero and the Y coordinates add to zero, apart from rounding. If the sums are noticeably off, check the origin, hole count, and rounding.
Existing parts do not always volunteer their PCD
Even numbered patterns are easier to measure than odd numbered patterns
With an even number of holes, one hole normally has another hole directly opposite it. Measure center to center across that pair and you have the BCD.
If you can only reach the hole edges, compensate for the hole diameter. An outside to outside measurement across opposite equal holes is the BCD plus one hole diameter. An inside to inside measurement is the BCD minus one hole diameter.
An odd numbered pattern has no directly opposite pair. You can measure the chord between neighboring centers and calculate:
BCD = adjacent chord ÷ sin(180° ÷ hole count)
Five lug wheel patterns are famous for creating awkward tape measure rituals. A template, calipers, manufacturer drawing, or coordinate measurement is usually more dependable than estimating the imaginary opposite point.
A loose bolt may fit several slightly different measurements. Measure the actual hole centers or use the specified pattern, not the position where a fastener happens to rest.
A correct pattern can still be mirrored
Confirm the origin, zero angle, and positive direction
This calculator uses the positive X direction, the right side of the circle, as zero degrees. Counterclockwise movement increases the angle. That is the conventional mathematical coordinate system.
Some machines, screens, drafting systems, and inspection reports use a different orientation. Screen coordinates often increase downward in Y. A rotary axis may define clockwise movement as positive. A drawing may place zero at the top. None of those systems is wrong, but mixing two of them can produce a beautifully accurate mirror image.
The center X and center Y fields let you shift the pattern into a larger machine coordinate system. If the part center is X 6.000 and Y 4.000, every hole coordinate is calculated from that origin instead of zero.
A numbered starting hole gives the angle table a physical reference. Without it, the same pattern can be rotated and still look mathematically correct.
Precision circles existed before digital readouts
Dividers, indexing plates, and dividing engines solved the same problem mechanically
Compasses and dividers have been used since antiquity to draw circles and transfer distances. A pair of dividers can step a chord around a circle, which is a physical version of dividing 360 degrees into equal parts.
Machinists later used indexing plates and dividing heads. A plate with carefully arranged hole circles allowed a spindle to rotate through repeatable fractions of a turn. Gears, splines, flutes, and bolt patterns could be divided without calculating every coordinate.
In 1775, English instrument maker Jesse Ramsden completed a circular dividing engine designed to divide arcs automatically. Machines of that kind helped instrument makers engrave precise scales for navigation, astronomy, surveying, and scientific measurement.
Rotary tables, numerical control, and CNC machining changed the motion, but not the geometry. The modern coordinate table is the same old divided circle expressed as numbers instead of scratches, pins, or index holes.
Fine angular scales demanded far more divisions than an ordinary bolt pattern, but both jobs begin with the same question: how do we divide one revolution evenly?
The formula is ideal; the part is employed in the real world
Practical drilling and inspection checks
- Locate the true part center before laying out any hole.
- Confirm whether the drawing gives BCD, radius, neighboring chord, or outside part diameter.
- Use one coordinate origin for every hole; do not step from the previous drilled hole.
- Check hole diameter, edge distance, countersink diameter, and counterbore diameter separately.
- Account for drill wander, spindle runout, backlash, fixture movement, and thermal expansion when the tolerance is tight.
- Spot or center drill before using a larger twist drill when the process allows it.
- Inspect at least one diameter, one neighboring chord, and the clocking angle of hole number one.
- Use the mating part or a verified template before committing an expensive flange or wheel.
The optional part diameter calculation gives a simple radial edge margin from the hole edge to the outside edge. It does not determine whether that margin is structurally adequate. Loads, material, fastener size, thread engagement, and the governing design requirements still matter.
Clearance helps assembly; it does not repair a pattern whose center, rotation, or diameter is wrong.
The circle has been divided fairly
Every hole has an angle, every center has coordinates, and none of them had to guess where they belonged.
You now know how BCD, angular step, chord spacing, coordinate origin, and rotation direction describe the same pattern.
That earns cheese fries. I arranged six of them around a plate, and yes, the adjacent chord is exactly the radius. Lunch can be educational without becoming unbearable.
Quick answers
Bolt circle questions
What is bolt circle diameter?
Bolt circle diameter, also called BCD or PCD, is the diameter of the imaginary circle passing through every hole center.
How do I calculate the angle between bolt circle holes?
Divide 360 degrees by the hole count. Six holes have a 60 degree step; five holes have a 72 degree step.
How do I calculate X and Y coordinates on a bolt circle?
Use X = center X + radius × cosine of the angle, and Y = center Y plus or minus radius × sine of the angle, depending on the chosen direction.
What is chord spacing on a bolt circle?
It is the straight distance between neighboring hole centers. The formula is 2 × radius × sin(180° ÷ hole count).
How can I measure an existing even numbered bolt pattern?
Measure center to center between opposite holes. That distance is the BCD.
Why are odd numbered bolt patterns harder to measure?
They have no directly opposite hole pair, so the BCD must be derived from a chord, coordinates, a template, or manufacturer data.
Does the calculator guarantee that a flange or wheel will fit?
No. It gives ideal geometry. Hole size, center accuracy, runout, clearance, edge distance, distortion, and tolerance still control the fit.