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Bolt Circle Calculator

Get the X and Y position of every hole in a bolt circle, ready to drill, from its diameter, number of holes, first-hole angle and center.

Level: Grades 10–11

  • Common Core standard HSG.SRT.C.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
  • Common Core standard HSF.TF.A.2Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Printable worksheet

Bolt Circle Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Units

The diameter of the circle through the hole centers (the pitch circle).

Measured from the +X axis (3 o'clock). Use 90 to put the first hole at 12 o'clock.

Number the holes

Leave the center at 0, 0 for coordinates relative to the circle's center.

Result

Center-to-center spacing of neighboring holes

2 in

6 holes on a 4 in circle are 60° apart, 2 in center to center.

Angle between holes
60 °
Bolt circle radius
2 in
Hole 1 X
2 in
Hole 1 Y
0 in

See the step-by-step solution ↓

Hole coordinates
Hole coordinates
HoleAngle (°)X (in)Y (in)
1020
26011.7321
3120-11.7321
4180-20
5240-1-1.7321
63001-1.7321

Step-by-step solution

  1. Radius: half the bolt circle diameter

    4 in2=2 in

  2. Angle between neighboring holes: a full turn divided by the number of holes

    360 °6=60 °

  3. Find hole 1's X position: start at the center, then add the radius times the cosine of its angle, the sideways part

    0+2×cos 0 °=2 in

  4. Find hole 1's Y position: start at the center, then add the radius times the sine of its angle, the up-and-down part

    0+2×sin 0 °=0 in

  5. Each next hole adds 60° counterclockwise, so hole 2 sits at 60°

    (x₂, y₂)=(1, 1.732051)

  6. Find the straight-line spacing between neighboring hole centers (the chord), the diameter times sin(180° ÷ n), so you can check it with calipers

    4×sin 30 °=2 in

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Get the exact X and Y position of every hole in a bolt circle, ready to drill. A bolt circle is a ring of evenly spaced holes on a circle of a set size. Flanges, wheel hubs, pipe fittings, motor mounts and fixture plates all use them.

To drill them on a mill with a digital readout or on a CNC machine, you need each hole's position as X and Y measurements from the center. That's a small trigonometry problem for every hole, and this calculator does the work for you.

Enter the bolt circle diameter, the number of holes and the angle of the first hole. If you like, add where the center sits on your part. You get the angle and position of every hole, numbered counterclockwise or clockwise. You also get the straight-line spacing between neighboring holes, so you can check a layout with calipers.

Where it's used

Why this math matters outside the classroom:

  • Drilling flanges. A machinist drills six holes on a 4 in bolt circle at (2, 0), (1, 1.7321) and so on, 60° apart.
  • Wheel hub patterns. A 5 × 114.3 mm hub pattern puts its holes 72° apart and 114.3 × sin 36°, about 67.18 mm, center to center.
  • CNC programming. A programmer turns a 4-hole pattern on a 100 mm circle at 45° into points at (±35.3553, ±35.3553).
  • Fixture plates. A toolmaker offsets a bolt circle center to (10, −5) so hole coordinates read straight off the part edge.
  • Checking a part. An inspector measures 2.0572 in between neighboring holes of a 5-hole pattern to confirm a 3.5 in bolt circle.

Grade level and standards: Grades 10–11; Common Core State Standards for Mathematics HSG.SRT.C.8, HSF.TF.A.2. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose inches or millimeters. The positions come out in the same unit.
  2. Enter the bolt circle diameter (measured through the hole centers) and the number of holes.
  3. Set the angle of the first hole from the +X axis, and the direction to number the rest.
  4. If you like, enter the center's X and Y position. Then read the table of hole positions.

What the inputs mean

Bolt circle diameter
This is the diameter of the circle that runs through the centers of the holes. It's sometimes called the pitch circle diameter (PCD). It isn't the outside diameter of the flange, or the size of the holes.
Angle of the first hole
Angles are measured from the +X axis, which points right (3 o'clock). Positive angles go counterclockwise, the usual math convention. Enter 90 for a first hole at 12 o'clock. Patterns are often turned so the holes sit evenly on either side of the vertical centerline. For 4 holes, that means starting at 45°.
Number the holes
The numbering direction only changes the order of the holes in the table, not where they are. The one difference: clockwise also steps the angles the other way from the first hole.
Center X
Leave both center values at 0 to get positions measured from the circle's center. That's how most machines zero a bolt circle. Enter the center's position on the part to get positions from the part's own zero instead.

Formula and method

r = D ÷ 2
angle step = 360° ÷ n
angle of hole k = start angle ± (k − 1) × angle step
x = center X + r × cos(angle)
y = center Y + r × sin(angle)
spacing between neighbors = D × sin(180° ÷ n)

D is the bolt circle diameter and r is its radius, half of D. n is the number of holes, and k is the number of one hole (1 for the first). The angle step is the turn from one hole to the next. x and y are a hole's position.

Every hole sits on the same circle. So each one is a point at distance r from the center, at its own angle. The cosine (cos) of the angle gives the sideways part of that distance, and the sine (sin) gives the up-and-down part.

Two neighboring holes and the center form a triangle with two equal sides of length r. The angle at the center is 360° ÷ n. Cut that triangle in half and you get a right triangle. So the distance between the holes is 2r × sin(180° ÷ n), which is D × sin(180° ÷ n).

Worked example

Six holes are spaced evenly on a 4 in bolt circle, with the first hole on the +X axis and the center at 0, 0.

  1. Radius: 4 ÷ 2 = 2 in. Angle step: 360 ÷ 6 = 60°.
  2. Hole 1 at 0°: x = 2 × cos 0° = 2, y = 2 × sin 0° = 0.
  3. Hole 2 at 60°: x = 2 × cos 60° = 1, y = 2 × sin 60° = 1.7321.
  4. Spacing between neighbors: 4 × sin 30° = 2 in.

Result: The holes sit at (2, 0), (1, 1.7321), (−1, 1.7321), (−2, 0), (−1, −1.7321) and (1, −1.7321), 2 in apart center to center.

Understanding the result

Use the table's positions with the machine zeroed on the circle's center. If your readout is zeroed on an edge of the part instead, enter the center's position, so the numbers are measured from that zero.

The neighbor spacing is a quick check. Measure between two neighboring holes with calipers (the outside distance minus one hole diameter) and compare.

Assumptions and limitations

  • The holes are evenly spaced around a full circle.
  • Angles follow the math convention: counterclockwise from the +X axis, looking at the side you're drilling.
  • It doesn't cover unevenly spaced patterns or partial arcs. For those, work out each angle yourself and use x = r cos θ, y = r sin θ, where θ is the hole's angle.
  • Positions are rounded to four decimal places in the table. That's finer than most manual machines read, but it may matter for very large circles.
  • It doesn't check whether the holes overlap or sit too close to the flange edge.

Frequently asked questions

How do I calculate bolt circle coordinates?

Divide 360° by the number of holes to get the angle step. For each hole, multiply the radius by the cosine of its angle for X, and by the sine for Y. Then add the center's position. On a 4 in circle, the hole at 60° is at (2 cos 60°, 2 sin 60°) = (1, 1.7321).

How do I measure the bolt circle diameter of an existing part?

For an even number of holes, measure center to center across two opposite holes. For an odd number, measure the spacing between two neighbors and divide by sin(180° ÷ n). With 5 holes, a 2.0572 in spacing gives a 3.5 in bolt circle.

What start angle should I use?

It depends on the drawing. Many patterns put a hole at 12 o'clock (90°). Others sit evenly on either side of the vertical centerline, which for n holes means starting at 90° − 180° ÷ n. A 4-hole pattern then starts at 45°.

Is the bolt circle the same as the pitch circle diameter?

Yes. Both names describe the circle through the hole centers. Wheel and hub patterns written like 5 × 114.3 mean 5 holes on a 114.3 mm bolt circle.

How this calculator is checked

Method: Polar-to-Cartesian coordinates on the bolt circle (x = cx + r cos θ, y = cy + r sin θ) with θ stepping by 360°/n; chord length D sin(180°/n). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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