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
Gear Span Measurement Calculator
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.
Span measurement explained simply
The span measurement is an inspection dimension: it is used to check whether the teeth of a gear have the right thickness. Two parallel measuring discs are placed across several teeth and the distance between them is measured. The calculator tells you what that reading should be and within which range a measured value is acceptable.
How it is measured
A gear tooth micrometer (a micrometer with disc-shaped anvils) is placed across k teeth. One disc touches the left flank of the first tooth, the other the right flank of the last tooth. The reading is the span Wk. Because the discs touch the flanks from the side, it does not matter whether the gear runs slightly out of true or its outside diameter deviates a little.
The inputs: what they mean and where they come from
| Quantity | What is it? | Where do I get the value? | What does it influence? |
|---|---|---|---|
| Normal module mn | The size of the teeth. The larger the module, the larger and stronger the teeth. The distance from tooth to tooth on the reference circle is π·m. | From the gear data table on the drawing (“module”). For an unknown spur gear without profile shift, roughly: outside diameter ÷ (number of teeth + 2), e.g. 128 mm ÷ 32 = 4 mm. Common values follow the module series (e.g. 1, 1.25, 1.5, 2, 2.5, 3, 4, 5 …). | The span grows almost in proportion to the module. A wrong module makes the whole result useless. |
| Number of teeth z | Number of teeth around the whole gear. | From the drawing, or simply count them on the gear. | More teeth mean a larger gear diameter, so the measurement spans more teeth and the span becomes larger. |
| Normal pressure angle αn | The flank angle of the tool that generates the teeth. It sets how steeply the tooth flanks are inclined. | Almost always 20° (standard value). Other values (e.g. 14.5° or 25°) are then stated explicitly on the drawing. | Sets the flank shape and therefore the span and the recommended number of teeth to span. |
| Helix angle β | How much the teeth are inclined to the gear axis. 0° means straight teeth (spur gear), otherwise a helical gear. | From the drawing. Enter 0 for straight teeth. | On a helical gear the number of teeth to span and the span change, and the gear must be wide enough for the discs to bridge the helix. |
| Profile shift coefficient x | States whether the teeth were deliberately made thicker (x > 0) or thinner (x < 0) during manufacture, for example to hit a given centre distance or to avoid undercut. | From the gear data table on the drawing (“profile shift x”, or as an amount x·m in mm, then divide by m). Usually 0 if not stated. | Shifts the span directly by 2·x·mn·sin αn. Example: x = 0.5 at mn = 4 mm makes Wk 1.37 mm larger. A forgotten profile shift is the most common reason for readings that seem wrong. |
| Teeth spanned k | How many teeth the measuring discs are placed across. | Enter 0 and the calculator picks a suitable value. If the drawing gives an inspection dimension such as “W4”, use the 4. | Each additional tooth makes Wk larger by one normal base pitch. An unsuitable k makes the discs touch too close to the tip or root and distorts the measurement. |
| Upper and lower tooth thickness allowance Asne / Asni | How much thinner than the theoretical size the teeth are meant to be. This is deliberate: it creates a little play (backlash) between two meshing gears so nothing jams. | From the gear data table on the drawing, negative on an external gear, e.g. −0.08 / −0.14 mm. They are chosen during design from the required backlash, e.g. to DIN 3967. Divide values in µm by 1,000. If not stated, leave both at 0 to get the nominal value only. | Set the limits within which the measured gear is acceptable. |
| Face width b | Width of the gear along its axis. | From the drawing, or measure it on the gear with a caliper. Only needed for helical gears, otherwise leave at 0. | Only checked: if the gear is narrower than the required minimum width, the span cannot be measured. |
The results: what they tell you and what you need them for
| Quantity | What does it tell me? | What do I need it for? |
|---|---|---|
| Span measurement Wk | The theoretical target (nominal value): the distance between the two parallel discs across k teeth if the teeth had exactly nominal thickness. | Basis of the inspection dimension on the drawing. For the actual check use the limits below: because of the negative allowances, a good gear always measures slightly below Wk. |
| Teeth spanned used / recommended | How many teeth were, or should be, spanned so the discs touch the flanks near mid tooth height. | Exactly this number of teeth must lie between the discs when measuring. It belongs on the drawing too (e.g. “span W4 over 4 teeth”). |
| Span Wk max and Wk min | Largest and smallest acceptable reading. In the example: 42.9353 mm and 42.8789 mm. | This is the actual check: a reading between Wk min and Wk max means the tooth thickness is right. Above Wk max the teeth are too thick (too little backlash, the gearbox may jam, usually reworkable). Below Wk min they are too thin (too much backlash, usually scrap). Measure at several points around the gear. |
| Span allowances Awe / Awi | The tooth thickness allowances converted to the span (Aw = As·cos αn). In the example −0.0752 / −0.1316 mm. | For the drawing: the inspection dimension is usually stated as nominal value with these allowances, e.g. “W4 = 43.011 −0.075/−0.132”. |
| Normal base pitch pbn | Distance from one flank to the corresponding flank of the next tooth, measured in the measuring direction. In the example 11.8085 mm. | Check value while measuring: if a reading is off by almost exactly this amount, one tooth too many or too few was spanned and the gear is not mismanufactured. |
| Normal base tooth thickness sbn | Thickness of one tooth at the base circle in the measuring direction. Wk = (k − 1)·pbn + sbn. | Cross-check of the calculation with the standard formula. Usually not needed on the shop floor. |
| Transverse pressure angle αt | Pressure angle measured perpendicular to the gear axis. Equal to αn for straight teeth, larger on a helical gear. | Intermediate value; also used in further gear calculations (e.g. centre distance). |
| Virtual tooth number zn | Tooth number of a notional spur gear that behaves like the helical gear during measurement. Equal to z for straight teeth. | Intermediate value from which the recommended number of teeth to span follows. |
| Base diameter db | Diameter of the notional circle from which the tooth flanks are “unwound” as involutes. The measuring discs lie tangent to this circle. | Geometric basis of the measurement; also used when inspecting the flank profile. |
| Required minimum face width | How wide a helical gear must at least be for the discs to bridge the inclined teeth at all. Zero for straight teeth. | Compare with the actual face width. If the gear is narrower, measure over balls or pins instead of the span. |
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.