How to use this calculator
- Enter the gear definition. Module (or DP), tooth count, pressure angle and profile shift from the drawing.
- Take the recommended span. Use the automatic span count so the anvil contact lands near the pitch circle.
- Measure across the flanks. Use flat anvils tangent to the flanks, clear of tip chamfers and root fillets; rock gently for the minimum reading.
- Compare against W. A reading below the theoretical W indicates thinner teeth (backlash allowance or wear); drawings usually tolerance W downward.
- Cross-check with one more tooth. Measure k+1 teeth too: the difference must equal the base pitch π·m·cos(α).
How it works
The span (base tangent) method measures gear tooth thickness with nothing
but calipers. When flat anvils contact two involute flanks spanning
k teeth, the measured line is tangent to the base circle, so
its length is a pure involute property:
W = m·cos(α) · [π·(k − 0.5) + z·inv(α)] + 2·x·m·sin(α)
with inv(α) = tan(α) − α. Nothing about the gear's outside
diameter, bore or runout enters the relation — which is exactly why span
measurement is the preferred shop check: a gear wobbling on its blank
still gives the same W.
Two useful structures fall out of the formula. First, each added tooth
adds exactly one base pitch p_b = π·m·cos(α). Second, profile
shift adds 2·x·m·sin(α), so the span verifies the shift a
gear was actually cut at — pair it with the
profile shift coefficient
calculator when the drawing calls one out. For gears too large or too
small to span, the
measurement-over-pins
method covers the same job with pins or balls.
Worked example
Verified against the live calculator
A 20-tooth spur gear, module 2 mm, 20° pressure angle, no profile shift.
The recommended span is k = round(20/9 + 0.5) = 3 teeth.
With inv(20°) = 0.014904:
W = 2 × cos 20° × [π × 2.5 + 20 × 0.014904] = 15.321 mm
The caliper contact sits at d_c = √(d_b² + W²) = 40.59 mm —
just outside the 40 mm pitch circle, comfortably below the 44 mm tip
circle. Spanning 4 teeth instead reads exactly one base pitch more:
15.321 + 5.904 = 21.225 mm. Cut the same gear with
x = +0.5 profile shift and every span grows by
2 × 0.5 × 2 × sin 20° = 0.684 mm.
Frequently asked questions
What is span measurement over teeth?
It is the distance W across k teeth measured with flat caliper or disc-micrometer anvils tangent to the base circle. Because the anvils touch two involute flanks, W = m·cos(α)·[π(k − 0.5) + z·inv(α)] + 2·x·m·sin(α), and the reading directly checks tooth thickness without depending on the gear OD or runout.
How many teeth should I measure across?
Pick k so the contact lands near the pitch circle: k ≈ z·α/180° + 0.5 (for 20°, roughly z/9 + 0.5, rounded). This calculator computes it automatically and flags spans whose contact diameter drifts toward the tip or root.
Why do successive spans differ by exactly one value?
Adding one more tooth to the span adds exactly one base pitch: W(k+1) − W(k) = π·m·cos(α). Measuring two adjacent spans and checking the difference against the base pitch is a quick self-test of both the gear and the measurement.
Does profile shift change the span measurement?
Yes — positive shift thickens the teeth, widening every span by 2·x·m·sin(α). Span measurement is the standard shop way to verify a gear was actually cut at the specified shift.
Can I use this for helical gears?
Not this page. Helical span measurement works in the normal plane with the normal module, helix-dependent virtual tooth count, and the anvils land diagonally across the face width. Use dedicated helical metrology references for those.
Method & assumptions
- External involute spur gears with an unmodified flank; standard full-depth proportions (ha* = 1) are assumed for the tip-diameter screen.
- The theoretical W is the zero-backlash value. Production drawings tolerance W downward (e.g., −0.05/−0.10 mm) to create backlash; compare against the drawing band, not the nominal alone.
- Anvils must contact on the involute, clear of tip chamfer and root fillet — the contact-diameter check screens for this but cannot see your specific tool relief.
- Helical gears, internal gears and gears with heavy undercut need their own metrology methods.