A bolt hole circle, evenly spaced holes around a center, is the friendliest piece of real trigonometry in CNC, because the math does exactly what you can see: sine and cosine place points on a circle, and a bolt pattern is points on a circle. Two formulas and a list of angles produce every coordinate, and once you have worked one set by hand the pattern is obvious, whether or not you let software do it next time.
The formula
For each hole, you need its angle and then its position:
- Angle of hole n = start angle + n times (360 / number of holes)
- X of hole n = center X + radius times cosine(angle)
- Y of hole n = center Y + radius times sine(angle)
That is the whole thing. The angles are evenly spaced (a 6-hole circle every 60 degrees, an 8-hole every 45), and sine and cosine convert each angle plus the radius into an X and Y offset from the center, the single clearest case of trigonometry earning its place in G-code.
A worked set
Take a 6-hole circle, 100 mm bolt-circle diameter (so radius 50), centered at X100 Y100, first hole at 0 degrees:
| Hole | Angle | X = 100 + 50 cos | Y = 100 + 50 sin |
|---|---|---|---|
| 1 | 0 | 150.000 | 100.000 |
| 2 | 60 | 125.000 | 143.301 |
| 3 | 120 | 75.000 | 143.301 |
| 4 | 180 | 50.000 | 100.000 |
| 5 | 240 | 75.000 | 56.699 |
| 6 | 300 | 125.000 | 56.699 |
Each row is the two formulas with the hole’s angle, and the symmetry (holes 2 and 6 share an X, 3 and 5 share an X) is a built-in check that the arithmetic is right. Feed those six positions to a drilling cycle and the pattern drills itself, each hole just an X Y under the modal cycle.
The three ways to do it in practice
You rarely have to grind the formula every time, but knowing it underpins all three options:
Hand-calculate with the formula, fine for simple patterns and the way to understand what is happening. A bolt-hole-circle canned cycle, if your control offers one: you supply the center, radius, hole count, and start angle, and the control runs the sine-and-cosine math internally, the canned-cycle convenience applied to patterns, with the exact format in your machine’s reference. Or CAM, which generates the points from a sketch for any pattern, simple or not. The formula remains worth knowing even when software does the work, because it lets you check the output and handle the cases, partial arcs, odd start angles, that a cycle may not cover.
Why this is the trig lesson worth doing
If any single exercise turns trig from abstract fear into concrete tool, it is working one bolt circle by hand. Sine and cosine stop being classroom symbols and become what they are in the shop: the functions that put holes where the print wants them on a circle. A beginner who computes one six-hole set understands CNC trig better than a lecture conveys, and from there the formula generalizes to any point-on-a-circle problem. The surrounding code, the drilling cycle, the coordinates, the modal behavior, is the standard core that the free 60-second rounds on the G-code practice page keep automatic, so when you program a bolt circle the only new thinking is the two formulas, and after the first one even those feel routine.
Sources
Frequently asked questions
How do you find the X and Y coordinates for a bolt hole circle?
With sine and cosine: X = center X + radius times cosine(angle), Y = center Y + radius times sine(angle), where each hole’s angle is the start angle + (360 / number of holes) times the hole index. Compute per hole for the coordinate list, or use a canned cycle or CAM.
What is the formula for evenly spaced holes on a circle?
Angle of hole n = start angle + n times (360 / number of holes); position X = center X + radius times cosine of that angle, Y = center Y + radius times sine. A 6-hole circle is every 60 degrees, an 8-hole every 45.
Do I have to calculate bolt circle coordinates by hand?
No: hand-calculate with the formula, use a bolt-hole-circle canned cycle if your control has one (give it center, radius, count, start angle), or let CAM generate the points. Knowing the formula lets you check the result and handle cases a cycle may not cover.
Why is the bolt circle a good way to learn CNC trig?
Because it is the clearest case of trig earning its place: sine and cosine place points on a circle, which is what a bolt pattern is, so the formula is concrete and immediately useful. Working one by hand teaches CNC trig better than a lecture.