Base Tangent Length
Compute span number k and base tangent length W for standard spur gears (involute function invα = tanα − α).
Inputs
Formula
invα = tanα − α; k = ceil(z/9); W = m·cosα·((k−0.5)π + z·invα)
Fundamentals
Base tangent length is the distance across $n$ teeth measured by caliper — an efficient gear inspection in mass production.
Base tangent over $n$ teeth $W=m[2.5(n-0.5)+0.014z]$ (standard spur, $\alpha=20^\circ$), $m$ module, $z$ teeth. Choose $n$ so caliper faces sit near the pitch circle.
History
Base-tangent measurement rests on involute geometry and is a simple check of composite gear error, grown with gear metrology.
Engineering applications
Used after hobbing or shaping to inspect tooth thickness and form in composite.
Glossary
| Base tangent $W$ | Common normal distance across $n$ teeth. |
| Teeth $n$ | Number spanned, sets measurement location. |
| Pitch circle | Where the tangent should sit. |
How to use
- Fill in Teeth z, Module m, Pressure angle α in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read Span k, Base tangent W, Involute invα in the results area.
Formula notesBase tangent length W_k = m·[(k−0.5)·cosα + z·invα], invα = tanα − α; k = span tooth count.
Formula · Worked Example · Knowledge
Formula
Base tangent length $W=m\cos\alpha((k-0.5)\pi+z\cdot\text{inv}\alpha)$, span $k=\lceil z/9\rceil$, involute $\text{inv}\alpha=\tan\alpha-\alpha$.
Worked Example
z=30, m=2, α=20° → k=4, invα≈0.0149, W≈21.54 mm.
Key Points
- Base tangent is a common gear inspection measure.
- Wrong span count misses the involute flank.
- Measured W back-solves actual tooth thickness and backlash.
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Teeth z | z | 30 | |
| Module m (mm) | m | mm | 2 |
| Pressure angle α (°) | alpha | ° | 20 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| Span k | k | |
| Base tangent W | W | mm |
| Involute invα | inv |
Applications
- Cutting parameters and speed
- Gear measurement (3-wire/base tangent)
- Blanking and deep-drawing process
FAQ
What formula does this tool use?
This tool computes per ISO / AGMA / ASME standard formulas: invα = tanα − α; k = ceil(z/9); W = m·cosα·((k−0.5)π + z·invα)
How accurate are the results?
Results match input precision, based on SI units and common engineering approximations; for critical duty re-check with a safety factor.
Where is it used?
Cutting parameters and speed