Layout and shop math
Hole location calculator: rows, grids and polar coordinates
X = X0 + i x s x cos(A), Y = Y0 + i x s x sin(A); five holes 0.750 in apart on a 30° line end at X 2.5981 in, Y 1.5000 in.
Technical review pending. Formulas and values are cited. This notice comes down after a machinist review.
| Hole | X (in) | Y (in) | X incremental (in) | Y incremental (in) |
|---|---|---|---|---|
| 1 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 2 | 0.6495 | 0.3750 | 0.6495 | 0.3750 |
| 3 | 1.2990 | 0.7500 | 0.6495 | 0.3750 |
| 4 | 1.9486 | 1.1250 | 0.6495 | 0.3750 |
| 5 | 2.5981 | 1.5000 | 0.6495 | 0.3750 |
Incremental = this hole minus the previous hole; hole 1 is measured from the start or reference point.
Holes 1 and 2, with your numbers:
X = X0 + i x s x cos(A) = 0 + 0 x 0.75 x cos(30°) = 0.0000 in
Y = Y0 + i x s x sin(A) = 0 + 0 x 0.75 x sin(30°) = 0.0000 in
X = X0 + i x s x cos(A) = 0 + 1 x 0.75 x cos(30°) = 0.6495 in
Y = Y0 + i x s x sin(A) = 0 + 1 x 0.75 x sin(30°) = 0.3750 in
Last hole (hole 5): X, Y
in
X = X0 + i x s x cos(A) = 0 + 4 x 0.75 x cos(30°) = 2.5981 in
Y = Y0 + i x s x sin(A) = 0 + 4 x 0.75 x sin(30°) = 1.5000 in. First to last: 4 x 0.75 = 3.0000 in along the line.
Step between holes: dX, dY
in
dX = s x cos(A) = 0.75 x cos(30°) = 0.6495 in
dY = s x sin(A) = 0.75 x sin(30°) = 0.3750 in. Key the absolute coordinates; rounded steps add up.
Source: Machinery's Handbook (hole coordinates); plane trigonometry and coordinate rotation as shown.
Coordinates are only as good as the zero they start from. Check the first hole and the last hole against the drawing before drilling.
How it works
Row of holes
X = X0 + i x s x cos(A); Y = Y0 + i x s x sin(A); s = L / (N - 1)
| Symbol | Meaning | Unit |
|---|---|---|
| X0, Y0 | First hole | in or mm |
| N | Number of holes | holes |
| i | Hole index from 0 (hole 1 is i = 0) | none |
| s | Center-to-center spacing | in or mm |
| L | First-to-last length | in or mm |
| A | Angle from +X, counterclockwise | ° |
Grid with rotation
u = (c - 1) x sx; v = (r - 1) x sy; X = X0 + u x cos(phi) - v x sin(phi); Y = Y0 + u x sin(phi) + v x cos(phi)
| Symbol | Meaning | Unit |
|---|---|---|
| R, C | Rows and columns | count |
| r, c | Row and column of a hole, from 1 | none |
| sx, sy | X and Y spacing before rotation | in or mm |
| u, v | Hole position in the unrotated grid | in or mm |
| phi | Grid rotation about hole 1, counterclockwise | ° |
Polar and reverse
X = Xr + R x cos(A); Y = Yr + R x sin(A); R = sqrt(dX^2 + dY^2); A = atan2(dY, dX)
| Symbol | Meaning | Unit |
|---|---|---|
| Xr, Yr | Reference point | in or mm |
| R | Radius from the reference | in or mm |
| dX, dY | Hole minus reference, X and Y | in or mm |
| A | Angle from +X, counterclockwise, shown 0 to 360° | ° |
Incremental columns come from the unrounded absolute values, then round for display. Holes spaced evenly on a circle come from the bolt circle calculator instead.
Worked example: row of 5 holes at 30°
5 holes, 0.750 in spacing, 30°, start X 0, Y 0.
- cos 30° = 0.866025; sin 30° = 0.500000. Step: dX = 0.750 x 0.866025 = 0.6495 in (0.649519); dY = 0.750 x 0.5 = 0.3750 in.
- Hole 1: X 0.0000, Y 0.0000 | incr 0.0000, 0.0000.
- Hole 2: X 0.6495, Y 0.3750 | incr 0.6495, 0.3750.
- Hole 3: X 1.2990, Y 0.7500 | incr 0.6495, 0.3750.
- Hole 4: X 1.9486, Y 1.1250 | incr 0.6495, 0.3750.
- Hole 5: X 2.5981, Y 1.5000 | incr 0.6495, 0.3750.
- Check: first to last is 4 x 0.750 = 3.000 in along the line; 3.000 x cos 30° = 2.5981 in, 3.000 x sin 30° = 1.5000 in.
- Four rounded increments of 0.6495 add to 2.5980, not 2.5981 (2.598076 raw); the gap already shows at hole 4, 1.9485 against 1.9486. Absolute coordinates are the safer DRO entry.
Result (in): hole 5 at X 2.5981, Y 1.5000.
Worked example: 3 x 4 grid rotated 15°, metric
3 rows x 4 columns, sx 20 mm, sy 25 mm, rotated 15° about hole 1 at X 0, Y 0, row by row.
- cos 15° = 0.965926; sin 15° = 0.258819. Along a row: dX = 20 x 0.965926 = 19.319 mm, dY = 20 x 0.258819 = 5.176 mm. Row start shift: -25 x 0.258819 = -6.470 mm in X, 25 x 0.965926 = 24.148 mm in Y.
- Hole 1 (r1 c1): 0.000, 0.000 | incr 0.000, 0.000.
- Hole 2 (r1 c2): 19.319, 5.176 | incr 19.319, 5.176.
- Hole 3 (r1 c3): 38.637, 10.353 | incr 19.319, 5.176.
- Hole 4 (r1 c4): 57.956, 15.529 | incr 19.319, 5.176.
- Hole 5 (r2 c1): -6.470, 24.148 | incr -64.426, 8.619.
- Hole 6 (r2 c2): 12.848, 29.325 | incr 19.319, 5.176.
- Hole 7 (r2 c3): 32.167, 34.501 | incr 19.319, 5.176.
- Hole 8 (r2 c4): 51.485, 39.677 | incr 19.319, 5.176.
- Hole 9 (r3 c1): -12.941, 48.296 | incr -64.426, 8.619.
- Hole 10 (r3 c2): 6.378, 53.473 | incr 19.319, 5.176.
- Hole 11 (r3 c3): 25.696, 58.649 | incr 19.319, 5.176.
- Hole 12 (r3 c4): 45.015, 63.825 | incr 19.319, 5.176.
- Check: hole 12 is u 60, v 50 from hole 1; sqrt(45.015^2 + 63.825^2) = 78.102 mm = sqrt(60^2 + 50^2), so the rotation kept the spacing.
Result (mm): hole 12 at X 45.015, Y 63.825.
Worked example: polar and reverse
R 2.000 in at 35° from X 0, Y 0; then a hole at X 1.2500, Y -0.7500 in back to radius and angle.
- Polar: X = 2.000 x 0.819152 = 1.6383 in; Y = 2.000 x 0.573576 = 1.1472 in.
- Reverse: R = sqrt(1.5625 + 0.5625) = sqrt(2.1250) = 1.4577 in.
- A = atan2(-0.7500, 1.2500) = -30.9638°, shown 329.0362° (329° 2' 10").
Result: X 1.6383, Y 1.1472 in; 1.4577 in at 329.0362°.
Shop notes
- Key absolute coordinates into the DRO or program where you can. Rounded increments add up; in the row example four increments of 0.6495 in land 0.0001 in short of hole 5.
- Set the DRO zero on hole 1 (or the reference) and keep it; every row in the table reads from that zero.
- On a manual mill, approach every hole from the same direction to keep backlash out of the handwheel dials; the DRO reads the table directly, the dials do not.
- Angles run counterclockwise from +X (3 o’clock), as on the bolt circle calculator. For a clockwise angle, enter it negative.
FAQ
How do you lay out holes in a line at an angle?
Step each hole by s x cos(A) in X and s x sin(A) in Y. At 0.750 in and 30° that is 0.6495 in and 0.3750 in per hole.
Where is the fifth hole of a 0.750 in, 30° row?
X 2.5981 in, Y 1.5000 in from hole 1: 3.000 in along the line.
How do you rotate a hole pattern?
X = X0 + u x cos(phi) - v x sin(phi); Y = Y0 + u x sin(phi) + v x cos(phi). Hole 2 of a 20 mm grid rotated 15° lands at X 19.319 mm, Y 5.176 mm.
Absolute or incremental for a DRO?
Absolute: each hole reads from one zero, so rounding cannot stack. Incremental columns are there to check the step.
How do you convert X and Y to radius and angle?
R = sqrt(dX^2 + dY^2) and A = atan2(dY, dX). X 1.2500 in, Y -0.7500 in is 1.4577 in at 329.0362°.
Holes on a circle?
Use the bolt circle calculator; it spaces N holes evenly on the circle.
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