This bolt circle calculator gives the X-Y coordinates of every hole on a bolt circle from its diameter, number of holes and first-hole angle. It also gives the center-to-center distance between adjacent holes. You can offset the center to match your part's origin, number the holes clockwise or counterclockwise, and space them over a partial arc. You can copy the whole coordinate table as CSV for a CNC program, spreadsheet or CAD file. Reverse mode works backward: measure the distance between two holes and it returns the bolt circle diameter.
A bolt circle is simply a set of points evenly spaced around a circle. Every coordinate comes from two trigonometry formulas, and the diagram updates as you type so you can check the pattern visually before you drill.
How to use the bolt circle calculator
- Enter the bolt circle diameter, also called the pitch circle diameter (PCD). This is the circle through the hole centers, not the outside of the part.
- Enter the number of holes.
- Set the first hole angle. 0° puts hole 1 at 3 o'clock, on the +X axis. 90° puts it at 12 o'clock. A common choice is to put the first hole on an axis, or half a step off it so the holes straddle the axis.
- Choose the direction the holes are numbered: counterclockwise (the math convention) or clockwise.
- Enter the center X and Y if the circle isn't at your origin, for example 3.000, 2.500 from the corner of a plate.
- Optionally enter an arc span below 360° for holes on a partial arc, and a hole diameter to check the web of material left between holes.
- Read the coordinates from the table or press Copy CSV. The first hole and first chord are shown in orange in the diagram.
Bolt circle formula
The holes on a bolt circle are points given in polar form, radius r = D ÷ 2 and angle θ, which you convert to rectangular coordinates. OpenStax Precalculus §8.3 gives the conversion as x = r cos θ and y = r sin θ. For N holes starting at angle A with the circle centered at (X₀, Y₀):
- Angle step B = 360° ÷ N (or arc ÷ (N − 1) for a partial arc)
- θᵢ = A + i × B for i = 0, 1, … N − 1 (use −B for clockwise)
- xᵢ = X₀ + (D ÷ 2) × cos θᵢ
- yᵢ = Y₀ + (D ÷ 2) × sin θᵢ
Distance between adjacent holes
Two neighbouring holes and the center form an isosceles triangle with two sides of length r and angle B between them. By the law of cosines, the chord between them satisfies c² = r² + r² − 2r² cos B. The half-angle identity sin(B ÷ 2) = √((1 − cos B) ÷ 2) simplifies this to c = D × sin(B ÷ 2). On a full circle that is c = D × sin(180° ÷ N), and working backward, D = c ÷ sin(180° ÷ N), which is what reverse mode uses.
Bolt hole spacing constants
Machinists often keep a table of chord constants: multiply the bolt circle diameter by the constant to get the center-to-center distance between adjacent holes, or divide a measured spacing by it to get the diameter.
| Holes (N) | Angle step | Chord constant | Spacing on 10.000 BCD | BCD from spacing of 1.000 |
|---|---|---|---|---|
| 3 | 120.00° | 0.86603 | 8.6603 | 1.1547 |
| 4 | 90.00° | 0.70711 | 7.0711 | 1.4142 |
| 5 | 72.00° | 0.58779 | 5.8779 | 1.7013 |
| 6 | 60.00° | 0.50 | 5.00 | 2.00 |
| 7 | 51.429° | 0.43388 | 4.3388 | 2.3048 |
| 8 | 45.00° | 0.38268 | 3.8268 | 2.6131 |
| 9 | 40.00° | 0.34202 | 3.4202 | 2.9238 |
| 10 | 36.00° | 0.30902 | 3.0902 | 3.2361 |
| 12 | 30.00° | 0.25882 | 2.5882 | 3.8637 |
| 16 | 22.50° | 0.19509 | 1.9509 | 5.1258 |
| 20 | 18.00° | 0.15643 | 1.5643 | 6.3925 |
| 24 | 15.00° | 0.13053 | 1.3053 | 7.6613 |
Worked examples
6 holes on a 100 mm PCD, first hole at 0°
The step is 60° and the radius is 50 mm. Hole 1 is at (50, 0). Hole 2 is at (50 cos 60°, 50 sin 60°) = (25.00, 43.301). Hole 3 is at (-25.00, 43.301), and so on around the circle. Adjacent holes are 100 × sin 30° = 50.000 mm apart. With six holes the chord always equals the radius, which is why a compass set to the radius can step off a hexagon.
5 holes on a 4.5 in circle, hole 1 at 12 o'clock
Start at 90° and step 72°. Hole 1 is at (0, 2.25). Hole 2, counterclockwise, is at (-2.1399, 0.6953). Hole 3 is at (-1.3225, -1.8203). Adjacent holes are 4.5 × sin 36° = 2.645 in apart.
Finding an unknown pattern
Suppose you measure 2.645 in center to center between two neighbouring holes of a 5-hole pattern. Then D = 2.645 ÷ sin 36° = 4.50 in. To measure a center-to-center distance with calipers, measure the inside edges of two same-size holes and add one hole diameter.
4 holes over a 90° arc on a 200 mm circle
With a partial arc, the step is 90 ÷ (4 − 1) = 30°. The holes sit at 0°, 30°, 60° and 90°. The coordinates are (100.00, 0.00), (86.60, 50.00), (50.00, 86.60) and (0.00, 100.00).
ASME B16.5 Class 150 flange bolt circles
Pipe flanges are the most common bolt circles in industry. We couldn't check the ASME B16.5 standard itself, because it is a paid standard. The values below are from two independent distributor charts that reprint its Class 150 table and agree with each other. For NPS 4 through 24, every bolt circle, bolt count and hole diameter also matches a public agency's drawing, East Valley Water District Standard Drawing W-113, a Class 150 table signed by a licensed engineer (it does not name B16.5). The ½ to 3½ rows rest on the distributor charts alone. Confirm against the standard, or the flange's own drawing, before you machine a mating part. The chord column is our calculation.
| NPS | Bolt circle | Bolts | Hole dia. | Adjacent spacing |
|---|---|---|---|---|
| ½ | 2.38 | 4 | ⅝ | 1.683 |
| ¾ | 2.75 | 4 | ⅝ | 1.945 |
| 1 | 3.12 | 4 | ⅝ | 2.206 |
| 1¼ | 3.50 | 4 | ⅝ | 2.475 |
| 1½ | 3.88 | 4 | ⅝ | 2.744 |
| 2 | 4.75 | 4 | ¾ | 3.359 |
| 2½ | 5.50 | 4 | ¾ | 3.889 |
| 3 | 6.00 | 4 | ¾ | 4.243 |
| 3½ | 7.00 | 8 | ¾ | 2.679 |
| 4 | 7.50 | 8 | ¾ | 2.870 |
| 5 | 8.50 | 8 | ⅞ | 3.253 |
| 6 | 9.50 | 8 | ⅞ | 3.635 |
| 8 | 11.75 | 8 | ⅞ | 4.497 |
| 10 | 14.25 | 12 | 1 | 3.688 |
| 12 | 17.00 | 12 | 1 | 4.400 |
| 14 | 18.75 | 12 | 1⅛ | 4.853 |
| 16 | 21.25 | 16 | 1⅛ | 4.146 |
| 18 | 22.75 | 16 | 1¼ | 4.438 |
| 20 | 25.00 | 20 | 1¼ | 3.911 |
| 24 | 29.50 | 20 | 1⅜ | 4.615 |
Piping practice usually orients flange bolt holes to straddle the centerlines, which is sometimes called "two-holing." With 4, 8, 12, 16 or 20 bolts, that means a first-hole angle of half a step (45°, 22.5°, 15°, 11.25° or 9°) so no hole sits on the vertical or horizontal axis. To reproduce that layout, enter half the angle step as the first hole angle.
Checking a bolt pattern before drilling
- Measure two ways. With an even number of holes, opposite holes should be exactly one bolt circle diameter apart, center to center. With any number of holes, adjacent holes should all be one chord apart.
- Check the ligament. Enter the hole diameter to see the edge-to-edge gap. If it is small or negative, the holes break into each other, so use fewer holes or a larger circle.
- Mind the origin. CNC programs usually work from a part corner or a fixture origin. Enter that offset as center X and Y so the numbers can be used directly.
- Watch the angle direction. Machine controls and CAD programs can differ on clockwise versus counterclockwise, and on where 0° is. Compare hole 2 in the table with your drawing before you run a program.
Common mistakes
- Using the outside diameter of the part instead of the circle through the hole centers.
- Measuring edge to edge without adding one hole diameter to get center to center.
- Measuring straight across a 5-hole pattern. Odd patterns have no opposite hole, so measure adjacent holes and use reverse mode.
- Entering degrees into a calculator set to radians when checking by hand. The tool here takes degrees.
- Rounding too early. Keep 4 decimal places in inches, or 3 in millimetres, until the final layout.
For converting drawing units, use the mm to inches converter. For right-triangle layout checks, use the 30-60-90 triangle calculator. For round concrete piers that carry anchor-bolt patterns, use the Sonotube concrete calculator.
Frequently asked questions
What is a bolt circle?
A bolt circle is the imaginary circle that runs through the centers of a ring of equally spaced holes, such as the bolt holes on a pipe flange, a wheel hub or an anchor-bolt pattern on a column base. It is described by its diameter, also called the bolt circle diameter (BCD) or pitch circle diameter (PCD), and by the number of holes.
How do you calculate bolt circle hole coordinates?
Divide 360° by the number of holes to get the angle step. For hole i, starting from 0, the angle is the start angle + i × step. Then x = center X + r × cos(angle) and y = center Y + r × sin(angle), where r is half the bolt circle diameter. These are the standard polar-to-rectangular formulas.
How do I find the bolt circle diameter from the distance between two holes?
For adjacent holes on a full circle of N holes, D = c ÷ sin(180° ÷ N), where c is the center-to-center distance. For example, 5 holes spaced 2.645 in apart sit on a 4.50 in circle. Use the reverse mode in the calculator.
What is the distance between bolt holes on a bolt circle?
The straight-line distance between adjacent holes is the chord: c = D × sin(180° ÷ N). For 4 holes that is D × 0.7071, for 6 holes it is D × 0.5, and for 8 holes it is D × 0.3827.
What does PCD mean?
PCD stands for pitch circle diameter. It means the same thing as bolt circle diameter: the diameter of the circle through the hole centers. It is not the outer diameter of the part or the diameter of the holes.
How do you measure a 5-hole bolt circle?
With an odd number of holes, no two holes are directly opposite each other, so you can't measure the diameter straight across. Measure center to center between two adjacent holes and use reverse mode, or measure center to center between a hole and one of the two farthest holes, which is D × sin(72°) = D × 0.9511 for 5 holes.
Which way is 0 degrees on a bolt circle?
This calculator follows the math convention: 0° points along +X (3 o'clock) and angles increase counterclockwise, so 90° is 12 o'clock. Choose clockwise if your drawing numbers holes that way. Check your CNC control's or CAD program's documentation for its angle convention.
Can I use this for holes on a partial arc?
Yes. Set the arc span to less than 360°, and the holes are spaced from the first-hole angle to the end of the arc, with holes at both ends. For example, 4 holes over 90° are 30° apart.
What bolt circle does a 4 inch Class 150 flange have?
According to published ASME B16.5 charts, a 4 in NPS Class 150 flange has 8 bolts on a 7.50 in bolt circle, with ¾ in holes. Adjacent holes are 7.50 × sin(22.5°) = 2.870 in apart, center to center.
How accurate are the coordinates?
The math is exact to floating-point precision, and you can show 2 to 5 decimal places. In practice, accuracy depends on your layout or machining. For a manual layout, check your work by measuring the adjacent chord and the across-center distance.
Sources & method
- OpenStax Precalculus 2e, §8.3 Polar Coordinates: converting polar to rectangular, x = r cos θ, y = r sin θ
- OpenStax Precalculus 2e, §8.2 Non-right Triangles: Law of Cosines
- OpenStax Precalculus 2e, §7.3 Double-Angle, Half-Angle and Reduction Formulas
- East Valley Water District (California), Standard Drawing W-113: Class 150 bolt and flange table, NPS 4–30 (Apr 2022)
- Northwest Fastener, ASME B16.5 Class 150 flanges: bolting pattern and bolt sizes (chart)
- API International, ASME B16.5 Class 150 flange dimensions (cross-check)
Results are estimates for general information. Found an error? It helps everyone — see our methodology.