Hole Pattern Calculator: Bolt Circle Math

A hole pattern calculator works out the X and Y coordinates for drilling or machining a circular arrangement of holes, usually called a bolt circle or pitch circle. You enter the circle diameter, the number of holes, and the angle of the first hole, and it returns the position of every hole relative to the center. This is the same math used for wheel bolt patterns, flanges, pipe covers, and any part where holes must be evenly spaced around a circle.

Formula

The position of each hole on a bolt circle is found with basic trigonometry. For hole number i (starting at 1):

X_i = R × cos(θ₁ + (i − 1) × Δθ)
Y_i = R × sin(θ₁ + (i − 1) × Δθ)

• R = radius of the bolt circle = circle diameter ÷ 2 (same unit as the diameter)
• θ₁ = angle of the first hole, measured counterclockwise from the positive X-axis (degrees)
• Δθ = angular step = 360° ÷ number of holes
• i = hole number, from 1 to n

All angles are in degrees. If your calculator uses radians, convert by multiplying degrees by π/180.

Worked Example

Using the default values: circle diameter = 100 mm, 6 holes, first hole at 0°.

1. Radius R = 100 ÷ 2 = 50 mm
2. Angle step Δθ = 360 ÷ 6 = 60°
3. Hole 1: θ = 0° → X = 50 × cos(0°) = 50.00, Y = 50 × sin(0°) = 0.00
4. Hole 2: θ = 60° → X = 50 × cos(60°) = 25.00, Y = 50 × sin(60°) = 43.30
5. Hole 3: θ = 120° → X = 50 × cos(120°) = −25.00, Y = 50 × sin(120°) = 43.30
6. Hole 4: θ = 180° → X = 50 × cos(180°) = −50.00, Y = 50 × sin(180°) = 0.00
7. Hole 5: θ = 240° → X = 50 × cos(240°) = −25.00, Y = 50 × sin(240°) = −43.30
8. Hole 6: θ = 300° → X = 50 × cos(300°) = 25.00, Y = 50 × sin(300°) = −43.30

The calculator gives the same coordinates: (50.00, 0.00), (25.00, 43.30), (−25.00, 43.30), (−50.00, 0.00), (−25.00, −43.30), (25.00, −43.30). All rounded to two decimals.

When to Use a Hole Pattern Calculator

This tool is for any job where holes must sit on a circle with equal angular spacing. Common uses include:

• Wheel bolt patterns (4-lug, 5-lug, 6-lug) for adapters or custom hubs
• Pipe flanges and gaskets with multiple bolt holes
• Loudspeaker driver mounting holes
• Lighting fixture or electrical box mounting patterns
• Machining a rotary table or CNC work
• Drilling hole patterns in wood, metal, or plastic

If you only need the bolt circle diameter (the distance across opposite holes), that is a different calculation. For an even number of holes, it is simply the distance between two opposite hole centers. For an odd number, use the chord formula: BCD = hole distance ÷ sin(180° ÷ n).

How to Read the Results

The coordinates are relative to the center of the circle. Positive X is to the right, positive Y is up (if you use the standard Cartesian convention). To drill, you would typically set your work zero at the center, then move to each (X, Y) position. If you are using a manual mill with a digital readout (DRO), you can enter these coordinates directly. For a drill press, you might instead mark the holes by measuring the chord distance between adjacent holes: chord = 2 × R × sin(180° ÷ n). For a 100 mm diameter, 6-hole circle, the chord is 2 × 50 × sin(30°) = 50 mm.

Common Mistakes and Accuracy Tips

• Mixing radius and diameter: Always halve the diameter before using the formula. Entering the diameter as the radius doubles all coordinates.
• Degree vs. radian mode: Make sure your calculator or software is in degree mode when using degrees. A common error is getting coordinates that look like a rotated pattern.
• Angle direction: The formula assumes counterclockwise from the positive X-axis. If your machine uses clockwise or a different zero angle, rotate the pattern by adjusting the first angle.
• Rounding too early: Round only the final coordinates, not intermediate values. For 5 holes, the sine and cosine of 72° are not nice decimals; rounding early can shift holes by a few hundredths of a millimeter.
• Hole count vs. hole number: The formula uses (i − 1) in the angle. If you start counting at 0, you can use i directly.

Limitations of the Formula

The math assumes holes are points on a perfect circle. It does not account for:

• Hole diameter – the coordinates are for the center of each hole, not the edge.
• Material thickness or drilling angle.
• Manufacturing tolerances – real parts need proper tolerances for bolt holes.
• Non-circular patterns (e.g., oval or rectangular bolt patterns).

For a 5-hole pattern, the coordinates are not symmetric in a simple way, so a calculator is especially helpful. The chord distance between adjacent holes is the same for all pairs, but the X and Y values are not obvious by eye.

Frequently Asked Questions

What is a bolt circle diameter?

The bolt circle diameter (BCD) is the diameter of the imaginary circle that passes through the centers of all holes in a circular pattern. It is often called the pitch circle diameter (PCD). For a wheel with 5 studs, the BCD is measured from the center of one stud to the center of the opposite (non-adjacent) stud, then multiplied by 1.701. For 4 studs, it is the distance between opposite studs.

How do I find the angle of the first hole?

The angle is measured from the positive X-axis (usually pointing to the right) counterclockwise. If you don’t care about orientation, set it to 0° and rotate the whole pattern as needed. Some machines use the Y-axis as zero; adjust accordingly.

Can I use this for a 3-hole pattern?

Yes, the formula works for any number of holes from 1 upward. For 3 holes, the angular step is 120°. The coordinates will be at 0°, 120°, and 240°.

What if my holes are not evenly spaced?

This calculator assumes equal angular spacing. For uneven patterns, you need to specify each hole’s angle individually. The same trigonometric formula applies per hole, but there is no single step value.

Does the hole diameter affect the coordinates?

No, the coordinates are for the center of each hole. The hole diameter only affects the physical drilling, not the position. You can drill any hole size at the computed center.

Why do I get negative coordinates?

Negative values are normal. They simply mean the hole is to the left of center (negative X) or below center (negative Y). If you prefer all positive coordinates, you can shift your work zero to a corner, but then you must add the same offset to every coordinate.

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in d | Circle diameter | mm | 100 | 0
in n | Number of holes | - | 6 | 3
in a | Angle of first hole | degrees | 0 | 0
out x | X coordinate | mm | (d/2) * cosd(a) | 2
out y | Y coordinate | mm | (d/2) * sind(a) | 2
out ang | Angle step | degrees | 360 / n | 1