Inputs
Normal module, number of teeth, normal pressure angle, helix angle and profile shift coefficient; optionally the number of teeth to span, the tooth thickness allowances and the face width.
Gears · ISO 21771-2
Span measurement over k teeth
Calculate the span measurement Wk of an external cylindrical gear – the inspection dimension actually used to check tooth thickness in production. The calculator picks a sensible number of teeth to span, converts the tooth thickness allowances into span allowances and returns the maximum and minimum for the check.
External cylindrical gears with unmodified flanks, spur or helical. Internal gears, profile modifications and measurement over balls or pins are not covered; for helical gears that are too narrow the calculator states that a span measurement is not possible.
WkISO 21771-2
02
Span measurement
Set the gear data and calculate.
Inputs and method
Span measurement to ISO 21771-2
Calculate the span measurement over k teeth and the limits that follow from the tooth thickness allowances.
Calculation
ISO 21771-2 states the span as Wk = (k−1)·pbn + sbn. Expanded into drawing quantities this becomes Wk = mn·cos αn·[π·(k−0.5) + z·inv αt] + 2·x·mn·sin αn with tan αt = tan αn/cos β. The number of teeth to span follows as k = zn·αn/180° + 0.5, rounded, with the virtual tooth number zn = z/cos³β. The limits come from the tooth thickness allowances via Aw = As·cos αn.
Example
mn = 4 mm, z = 30, αn = 20°, x = 0 give k = 4 and Wk = 43.0105 mm. One normal base pitch is 11.8085 mm – exactly the amount by which the dimension changes when one more tooth is spanned.
Sources and limits: ISO 21771-2:2025 formula (77) and formulae (78) and (79) for the limits; the expanded form and the number of teeth to span per Roloff/Matek, Maschinenelemente Formelsammlung, 15th edition, section 21, no. 80 and 81. Valid for external cylindrical gears with unmodified flanks; internal gears, profile modifications and measurement over balls or pins are not covered.
Span measurement to ISO 21771-2 in detail
Calculate the span measurement over k teeth and the limits that follow from the tooth thickness allowances.
What is the span measurement?
The span measurement is the distance between two parallel measuring faces that contact one right and one left flank across several teeth. Because both faces stand tangent to the base circle, what is really measured is a length along the line of action: (k−1) base pitches plus one tooth thickness on the base circle. That is exactly why the span is the most robust check of tooth thickness obtainable with a simple micrometer.
Formula and variables
Wk = mn · cos αn · [π · (k − 0.5) + z · inv αt] + 2 · x · mn · sin αn
Standard form per ISO 21771-2: Wk = (k − 1) · pbn + sbntan αt = tan αn / cos β · inv α = tan α − αTeeth spanned: k = zn · αn/180° + 0.5 (rounded), zn = z/cos³βLimits: Wk max/min = Wk + As · cos αn
| Symbol / input | Meaning |
|---|---|
| mn, z, x | Normal module, number of teeth and profile shift coefficient of the gear. |
| αn, αt, β | Normal and transverse pressure angle and the helix angle. |
| k, pbn | Teeth spanned and the normal base pitch, by which Wk changes per tooth. |
| Asne, Asni | Upper and lower tooth thickness allowance, negative on an external gear. |
Choose the inputs correctly
Normal module, number of teeth, normal pressure angle, helix angle and profile shift coefficient; optionally the number of teeth to span, the tooth thickness allowances and the face width.
How to use the calculator
Take the normal module, number of teeth, pressure angle, helix angle and profile shift coefficient from the gear drawing. Leave the number of teeth to span at 0 so the calculator selects a suitable value. Then enter the tooth thickness allowances – negative on an external gear – to obtain the maximum and minimum; for a helical gear add the face width so the calculator can check whether the measurement is possible at all.
Worked example
mn = 4 mm, z = 30, αn = 20°, x = 0 give k = 4 and Wk = 43.0105 mm. One normal base pitch is 11.8085 mm – exactly the amount by which the dimension changes when one more tooth is spanned.
How should the result be used?
The nominal value alone is not yet an inspection statement – only the limits from the tooth thickness allowances say whether a measured gear is acceptable. Between them lies the entire permissible manufacturing scatter. The normal base pitch is the practical check value: if a reading differs by exactly that amount, one tooth too many or too few was spanned rather than the gear being mismanufactured.
Module, span and allowances in millimetres, angles in degrees. Tooth numbers, teeth spanned and the profile shift coefficient are dimensionless. Drawings often give tooth thickness allowances in micrometres – divide by 1,000 in that case.
Typical applications
Production and incoming inspection of cylindrical gears, defining the inspection dimension on the gear drawing, checking the result after hobbing or shaping, and determining the remaining tooth thickness on a used gear.
Assumptions, limits and common mistakes
ISO 21771-2:2025 formula (77) and formulae (78) and (79) for the limits; the expanded form and the number of teeth to span per Roloff/Matek, Maschinenelemente Formelsammlung, 15th edition, section 21, no. 80 and 81. Valid for external cylindrical gears with unmodified flanks; internal gears, profile modifications and measurement over balls or pins are not covered.
Common mistake: Do not forget the profile shift: it moves the span by 2·x·mn·sin αn and is the most common cause of an apparently wrong dimension. On a helical gear do not enter the transverse module instead of the normal module. Enter the tooth thickness allowances as negative on an external gear – positive values would reverse the backlash. And do not pick the number of teeth to span freely without checking where the anvils then contact.
Frequently asked questions
Why measure across several teeth instead of the tooth thickness directly?
Because the anvils lie tangent to the base circle and touch the involute there. That makes the dimension independent of the tip and root diameters and – crucially – independent of runout in the mounting. A direct tooth thickness measurement would have to start from a reference diameter whose own position carries error.
How is the number of teeth to span chosen?
So the anvils touch the flanks near mid tooth height, where the involute is unmodified and the measurement is least sensitive. The guide value k = zn·αn/180° + 0.5 achieves exactly that. Changing k by one always changes the span by exactly one normal base pitch, which is also a good check at the instrument.
Why are the tooth thickness allowances negative?
Because a gear is deliberately made with teeth thinner than nominal so that backlash exists in mesh. Without it the gearing would jam as soon as temperature, centre distance or lubricant film changed. The allowances act on the span through the factor cos αn, so they are slightly smaller there in magnitude.
When can the span not be measured?
When the gear is too narrow for its helix angle: the anvils have to span the helix, which needs at least Wk·sin βb of face width. On narrow, strongly helical gears that condition is violated and the measurement has to be taken over balls or pins instead – a method this calculator deliberately does not cover.
Does this apply to internal gears?
No. On internal gears the sign convention of the allowances reverses, and ISO 21771-2 gives separate formulae in which the tooth thickness is added rather than subtracted. This calculator is explicitly limited to external gears.