Base Tangent Length

Calculate the span tooth count and base tangent length for standard spur gears, with optional backlash correction

Inputs

Z

Number of Teeth

\alpha

Pressure Angle

Typically 20° or 14.5° for standard gears

m

Module

Formula Interpretation

Span Teeth Count

n = \dfrac{\alpha \cdot Z}{180} + 0.5

The number of teeth to span $n$ is chosen so the micrometer anvils contact the tooth flanks near the mid-height of the involute. Rounding to the nearest integer gives the standard span.

Standard Base Tangent

E_{n0} = m\cos\alpha\!\left[\pi(n - 0.5) + Z \cdot \mathrm{inv}(\alpha)\right]

The general formula uses the involute function $\mathrm{inv}(\alpha)$ . For standard gears at α=20° and α=14.5° the tabulated coefficients reduce the formula to a linear expression in $Z$ , $n$ and $m$ .

Backlash Cut-in & Correction

\Delta t = \dfrac{B}{2\sin\alpha}

To produce gear-pair backlash $B$ , each gear is cut deeper by $\Delta t$ . The base tangent length is reduced by exactly $B$ , making it easy to verify the cut depth during machining.

Knowledge Points

Base Tangent Measurement Principle

A base tangent micrometer spans n teeth and measures the common normal (tangent to the base circle) across them. This length equals the sum of n base pitches minus one tooth thickness, expressed in the general formula E_n0 = m·cosα·[π(n−0.5) + Z·inv(α)]. It is independent of eccentricity errors and is therefore a reliable accuracy check.

Involute Function inv(α)

The involute function inv(α) = tan(α) − α (α in radians) appears in all involute gear geometry. For α=20°: inv(20°) ≈ 0.01490. For α=14.5°: inv(14.5°) ≈ 0.00540. The involute function maps the pressure angle to the tooth form and determines how the base tangent length changes with the number of teeth.

Backlash and Cut-in Depth

Backlash B is the tangential play between mating teeth. To introduce backlash, each gear is cut slightly deeper than standard by Δt = B/(2sinα). The resulting reduction in base tangent length is exactly B, giving a direct in-process measurement during hobbing or shaping: simply measure E_n and stop when it reaches E_n0 − B.

Worked Example

A gear pair with $\alpha = 20°$ , $m = 6\,\text{mm}$ , $Z_1 = 17$ , $Z_2 = 70$ is assembled at standard centre distance with backlash $B = 0.04m = 0.24\,\text{mm}$ . Since Z₁=17 is the undercut limit for 20°, the large gear is cut deeper to provide the required backlash.

Small gear (Z₁ = 17) — span count and standard base tangent

n = \frac{20 \times 17}{180} + 0.5 = 2.39 \approx 2

E_{n0} = (0.01401 \times 17 + 2.95213 \times 2 - 1.47606) \times 6 = 27.998\,\text{mm}

Large gear (Z₂ = 70) — span count, standard, then backlash-adjusted

n = \frac{20 \times 70}{180} + 0.5 = 8.28 \approx 8

E_{n0} = (0.01401 \times 70 + 2.95213 \times 8 - 1.47606) \times 6 = 138.730\,\text{mm}

\Delta t = \frac{0.24}{2\sin 20°} = \frac{0.24}{0.684} = 0.351\,\text{mm}

E_n = 138.730 - 0.24 = 138.490\,\text{mm}

Result: Small gear E_n0 = 27.998 mm (no extra cut needed); Large gear E_n = 138.490 mm (cut in by Δt = 0.351 mm to provide B = 0.24 mm backlash).

Extended Knowledge

•During hobbing or shaping, the operator measures the base tangent length continuously and reduces the radial infeed until E_n reaches the target value E_n0 (or E_n for a backlash-corrected gear). This avoids the need to stop the machine and check with a separate gauge.
•Profile-shifted gears have a modified base tangent formula: E_n = m·cosα·[π(n−0.5) + (Z + 2x·tanα)·inv(α)] where x is the profile shift coefficient. The span count n is also recalculated using the shifted pressure angle at the pitch circle.
•For helical gears, the normal base tangent length is measured in the normal plane with normal module m_n and normal pressure angle α_n. The formula is the same as for spur gears but uses normal-plane parameters throughout.

Related Standards & Articles

Authoritative references for the formulas used in this calculator

Standard

Gear Design Symbol Reference (GB/T 2821 / ISO 701)

A complete notation reference for gear design calculations — geometric parameters, load factors, strength properties, and safety factors — per GB/T 2821-2003 and ISO 701:1998, with full LaTeX symbol rendering.