CNC / machinist calculator
Gear Span Measurement Calculator (Base Tangent Length)
Span measurement, or base tangent length, is how you check tooth thickness on a gear with a plate or disc micrometer: you span a set number of teeth and compare the reading to a target. This calculator gives you both numbers, the ideal number of teeth to span so the anvils sit near the pitch line, and the exact span dimension to hit, for external spur and helical gears in module or diametral pitch, with profile shift and helix angle. It is the number your inspection sheet needs when cutting gears.
How it works
The span W is the length of the line tangent to the base circle that touches the same flank on the first and last of k teeth. Because that line lies on the base circle, the reading does not depend on where the micrometer sits radially, which is what makes span measurement repeatable. The value is W = m times cos of the pressure angle, times the quantity pi times k minus a half plus z times the involute of the pressure angle, plus a profile-shift term of two x m sin of the pressure angle.
The number of teeth to span, k, is chosen so the anvils contact the flanks near the reference diameter; too few or too many teeth and they touch near the tip or root and the reading drifts. The calculator rounds the standard formula to the nearest whole number of teeth, and you can override it if your inspection sheet calls for a specific span.
For a helical gear the span is measured in the normal plane, so the normal module and normal pressure angle are used and the helix folds into an equivalent tooth count. For a diametral-pitch gear the module is 25.4 divided by the DP. The formulas here are corroborated across KHK, DIN 3960 / ISO 21771, and reproduce a published helical worked example to four decimals.
Worked example
A module 2, 20-tooth, 20-degree spur gear: span 3 teeth, and the target base tangent length is 15.3209 mm (0.6032 in). Check that reading with a disc micrometer across three teeth and the tooth thickness is on size.
Frequently asked questions
What is span measurement or base tangent length?
It is the dimension across a number of gear teeth measured with a flat-anvil disc micrometer. Because it is taken along a line tangent to the base circle, it gives a repeatable check of tooth thickness independent of the exact micrometer position.
How many teeth should I span?
Enough that the micrometer anvils contact the flanks near the pitch diameter. The calculator computes the standard number from the tooth count and pressure angle and rounds it, but you can override it if a drawing or inspection sheet specifies a particular number of teeth.
Does this work for helical gears?
Yes. Enter the normal module and normal pressure angle and the helix angle, and the span is computed in the normal plane using the equivalent tooth count, which is how a helical gear is actually measured over teeth.
How do I use it for a diametral pitch gear?
Pick diametral pitch and enter the DP. The calculator converts to module as 25.4 divided by the DP, computes the span in millimetres, and also reports it in inches so you can work in whichever unit your micrometer reads.
What pressure angle should I enter?
Use the gear's pressure angle, most commonly 20 degrees, sometimes 14.5 or 25. It must match the actual gear, because the span depends directly on the pressure angle through the involute function and the cosine term.
Related calculators
Coverage
Implements Involute gear span measurement (DIN 3960 / ISO 21771 method): External spur and helical gears, Module or diametral pitch, any pressure angle, profile shift, Computed or manual number of teeth to span. Deliberately not offered yet:
- Internal (ring) gear span - internal gears are inspected over pins/balls rather than by a tooth span; a different measurement (revisit: with the over-pins calculator that covers internal gears)
- Tooth thickness allowance and backlash tolerances - the span here is the nominal theoretical value; the minus-allowance for backlash comes from the gear's tolerance class
Sources
- Tooth thickness and base tangent length (KHK Gears)
- Zahnweite (base tangent length), formula and worked example (Wikipedia)
- Elements of Metric Gear Technology, section 10.2 span measurement (QTC Gears mirror, PDF)
Every formula on this page is shown and sourced. See how we verify.
These calculators are for planning and as a starting point. Recommended speeds and feeds are published starting values that vary with your specific tool, coating, machine rigidity, workholding and coolant. Always start conservative, listen to the cut, and follow your tool maker data sheet.