How to use this calculator
- Pick the mode. Divisions for gear teeth, flutes and bolt circles; angle (deg/min/sec) for layout and angular features.
- Read turns and circle. Whole crank turns plus holes on the named circle — the smallest exact circle is chosen automatically.
- Set the sector arms. Span the hole count beyond the starting hole (the start hole is zero, not one) and pin the crank to the circle.
- Check the error output. Zero means the circle divides exactly. Anything else repeats every index — switch plates, use differential indexing or a CNC axis for exact work.
How it works
A dividing head is a worm drive with a vernier made of holes: the crank turns R times per spindle revolution, so every index is a fraction with denominator R —
turns = R / N (divisions) · turns = R × angle / 360 (angular) · holes = fraction × circle
The whole-number part is counted; the fractional part is walked out on a plate circle that turns it into whole holes. At 40:1 each crank turn is 9°, which is why so much machine-shop arithmetic reduces to ninths. The same hole-circle layout arithmetic for drilled features lives in the bolt circle calculator, the tooth counts being indexed usually come from the involute gear calculator, and decimal-degree ⇄ DMS conversion is one keystroke in the machinist calculator.
Worked example
Verified against the live calculator
Cutting a 27-tooth gear on a standard 40:1
head with B&S plates:
turns = 40/27 = 1 + 13/27 → 1 full turn + 13 holes on the 27 circle, every tooth
Exact — the 27 circle's holes divide the fraction with nothing left
over, so tooth 27 closes precisely on tooth 1. Compare
51 teeth: no standard circle divides 40/51, the nearest
setting (29 holes on 37) runs 17 arc-seconds off
per tooth, and the misclose stacks all the way around —
that job belongs to differential indexing. An angular example:
22°30′ at 40:1 is 2.5 turns = 2 turns + 8 holes on the
16 circle; on a 90:1 rotary table, 5° is
1¼ turns = 1 turn + 4 holes on the same circle.
Frequently asked questions
How do you calculate dividing head indexing?
Turns of the crank = 40 ÷ N on a standard 40:1 head. For 27 divisions that is 40/27 = 1 turn plus 13/27 — one full crank turn plus 13 holes on the 27-hole circle. The whole-number part is counted, the fraction is made on a plate circle the denominator divides.
What are the standard dividing head plate hole circles?
The Brown & Sharpe three-plate set: 15-16-17-18-19-20, 21-23-27-29-31-33 and 37-39-41-43-47-49 holes. Cincinnati heads use one large double-sided plate with different counts, and import heads vary — the calculator takes any circle as a custom entry.
How do you index an angle in degrees, minutes and seconds?
One crank turn is 360/R degrees — 9° on a 40:1 head. Turns = angle ÷ 9 (in degrees): 22°30′ needs 2.5 turns, made as 2 turns plus 8 holes on the 16 circle. Angles that are not multiples of small fractions of 9° may not land exactly on any circle — the error output shows the residual in arc-seconds.
What divisions can a 40:1 dividing head NOT do with standard plates?
Any N whose reduced fraction 40/N needs a hole circle the set lacks — the classic examples are primes and odd counts above 49, like 51, 53 or 59. For 51 the nearest setting (29 holes on 37) is 17 arc-seconds off per index, and the error repeats every tooth. Those counts need differential indexing with change gears, or a CNC rotary axis.
Method & assumptions
- Plain (simple) and angular indexing on a single worm: turns = R/N or R·angle/360. Differential indexing (change gears driving the plate) and compound indexing are not computed — the error output tells you when you need them.
- Standard Brown & Sharpe three-plate hole set verified against the Navy machinery repair manual and machining references; Cincinnati/import plates differ and enter as a custom circle.
- Holes are hole spaces beyond the starting hole — sector arms count spaces, the start hole is zero.
- Backlash discipline is yours: always approach the pin in the same crank direction, and back off past the hole and return if you overshoot.