Where do the holes go?

What the numbers are for

A bolt circle (the pitch circle diameter, PCD) is the circle that passes through the centers of a ring of equally spaced holes: the flange of a pipe, a wheel hub, a pulley, a bearing cap or a round cover plate. Whoever makes the part needs the position of every hole, either as X and Y from a datum for the mill or the CNC, or as the straight distance between neighbors for a divider and a caliper. The calculator gives both from the number of holes and the diameter of the circle, and it also works the other way, finding the diameter from a measured distance between holes. In row mode it spaces a number of holes evenly between two points or at a given pitch, at any angle.

The formulas

For n holes, circle radius r = D ÷ 2, and a first hole at angle θ₀ from the +X axis, hole k sits at x = cx + r · cos(θ₀ + k · 360°/n) and y = cy + r · sin(θ₀ + k · 360°/n). The angle between neighbors is 360° ÷ n. Two neighboring holes and the center form an isosceles triangle, so the straight chord between hole centers is D · sin(180° ÷ n), and the diameter from a measured chord is chord ÷ sin(180° ÷ n). In a row of n holes the pitch is the length from the first to the last hole divided by n − 1, the number of spaces.

Worked examples

Six holes on a 4 inch circle. The chord is 4 × sin 30° = 2 inches, the same as the radius, and the holes are at (2.000, 0.000), (1.000, 1.732), (−1.000, 1.732), (−2.000, 0.000), (−1.000, −1.732), (1.000, −1.732). A 5 × 114.3 mm wheel pattern. Five studs on a 114.3 mm circle are 67.18 mm apart (114.3 × sin 36°); measured the other way, a chord of 67.18 mm gives a circle of 114.30 mm. Eight holes on a 100 mm circle, first hole at 22.5°. The chord is 38.268 mm and hole 1 is at (46.194, 19.134). A row. Seven holes between points 18 inches apart have a pitch of 18 ÷ 6 = 3.0000 inches.

Limits and common mistakes

Counting holes instead of spaces in a row is the classic error: seven holes have six spaces, so the pitch is the length divided by six, not by seven. For a circle, the reverse: the number of spaces equals the number of holes. The values here are nominal: the position tolerance on the drawing (often given as one tolerance zone for the whole group of holes) and the machine's accuracy decide what you get. A chord measured from the edges of two holes, instead of from their centers, is wrong by one hole diameter: measure edge to edge on the near sides of two holes of the same size and add the hole diameter. Finally, check the material left between holes: the optional hole diameter shows it, and a web of zero or less means the holes break into each other.

How do I calculate bolt circle coordinates?

For n holes on a circle of diameter D with the center at (cx, cy), hole k (k = 0 … n − 1) is at x = cx + (D/2) cos(θ₀ + k × 360°/n) and y = cy + (D/2) sin(θ₀ + k × 360°/n), where θ₀ is the angle of the first hole measured counterclockwise from the +X axis. For 6 holes on a 4 inch circle the holes are at (2.000, 0.000), (1.000, 1.732), (−1.000, 1.732), (−2.000, 0.000), (−1.000, −1.732), (1.000, −1.732).

What is the distance between holes on a bolt circle?

The straight-line distance from one hole center to the next (the chord) is D × sin(180° ÷ n). For six holes it is exactly the radius, 2 inches on a 4 inch circle; for five holes on a 114.3 mm circle (the 5 × 114.3 wheel pattern) it is 67.18 mm. The arc between holes, π × D ÷ n, is longer: 71.82 mm in that example. Machinists use the chord to set a divider or to check a template with calipers.

How do I find the diameter of a bolt circle I can only partly measure?

With an even number of holes, measure center to center across the circle. With an odd number you cannot measure across, so measure the distance between two neighboring holes (the chord) and divide by sin(180° ÷ n): for five holes the diameter is the chord × 1.7013; a chord of 67.18 mm gives 114.30 mm. Choose “spacing between holes” in the form and the calculator does this division.

Which direction is the angle measured?

Counterclockwise from the positive X axis (to the right), as in school trigonometry and in most CNC controllers, with Y pointing up. Hole 1 is at the start angle and the others follow counterclockwise. If your drawing counts clockwise, enter 360° minus the angle, or mirror the Y values.

How many decimals do I need?

Four decimals of an inch (0.0001 in) or three decimals of a millimeter (0.001 mm) are what a CNC program normally carries, and they are what the table shows. The tolerance on the drawing, not the decimals, decides what you can hold: the position of the holes in a pattern is usually toleranced for the whole group, so the coordinates are the nominal values.

Does this work for woodworking, jigs and shelf pins?

Yes: the arithmetic is the same for a drilling template, a circular table-top fastener pattern, a row of shelf-pin holes or the holes in a pegboard. For a row, enter the number of holes and the distance from the first to the last (or the pitch) and an angle for a sloped row. Set the optional hole diameter to see how much material is left between neighboring holes.

Sources: plane trigonometry (coordinates on a circle; chord = D · sin(180°/n)); the chord factors for a circle of diameter 1 in the table above are sin(180°/n), computed here (n = 5: 0.58779, n = 7: 0.43388, n = 12: 0.25882); wheel-pattern notation (number of studs × circle diameter) as used in automotive references. Related: polygon calculator, circle calculator, right triangle calculator, tap drill calculator and bend allowance calculator. How we check our calculators.