Involute Function Calculator — Gear Design
Compute the involute function inv(α) = tan(α) − α for any pressure angle, with optional gear parameters: base circle radius, module, addendum and dedendum for standard spur gears.
°
mm
inv(α) = tan(α) − α (α in radians); dimensionless
- 1
Convert α to radians
20 × π ÷ 180 = 0.349066 - 2
Compute tan(α)
tan(0.349066) = 0.36397 - 3
Involute inv(α) = tan(α) − α
0.36397 − 0.349066 = 0.014904α must be in radians; the result is dimensionless.
How does this calculator work?
inv(α) = tan(α) − α (α in radians) is the gear involute function — dimensionless, monotonically increasing, used in gear tooth thickness and centre-distance calculations. Standard values: inv(14.5°) ≈ 0.00546, inv(20°) ≈ 0.01490. The base circle r_b = r_p × cos(α); module m = 2r_p / z. Addendum = 1m, dedendum = 1.25m per ISO 54.
Formula
How this is calculated
The involute of a circle is the curve traced by the end of a taut string unwound from the circle. In gear engineering, involute profiles are used for tooth flanks because two meshing gears with involute teeth maintain a constant velocity ratio regardless of small variations in centre distance — a crucial property for smooth transmission.
The involute function inv(α) = tan(α) − α arises in the relationship between the pressure angle α on the pitch circle and the tooth thickness at other radii. It is dimensionless and monotonically increasing: as α increases from 0°, inv(α) grows from zero. Standard gear pressure angles are 14.5° (older designs) and 20° (most modern gears), giving inv values of approximately 0.00546 and 0.01490 respectively.
For a spur gear with pitch circle radius r_p and z teeth, the module m = 2r_p / z (mm) defines the scale of the tooth profile. The base circle, at radius r_b = r_p × cos(α), is the circle from which the involute tooth profile is generated. The addendum (tooth tip height above pitch circle) is 1 module for a standard gear, and the dedendum (root depth below pitch circle) is 1.25 modules. These standard proportions follow ISO 54 / DIN 867 (2025-edition values, applicable to most general-purpose gearing).
Frequently asked questions
The involute function appears in equations for tooth thickness at any radius, the working pressure angle of a gear pair, and backlash calculations. Given a desired tooth thickness at the pitch circle, you solve for α using inv(α) — typically by numerical iteration — to find the correct profile shift or centre distance.
14.5° was the original standard derived from early rack-and-pinion designs; it allows quieter operation at light loads. 20° became the modern standard because it gives stronger, less-undercutting tooth profiles and is now preferred for most industrial gearing. 25° is used in some heavy-duty applications.
When a gear has too few teeth, the tool that cuts the tooth profile removes material from the tooth base (undercutting), weakening it. A higher pressure angle and a larger module both reduce the minimum number of teeth before undercutting occurs. For 20° pressure angle the minimum is typically 17 teeth for a standard gear.
Also known as
TG we-Calculate Editorial Team. (2026). Involute Function Calculator — Gear Design [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/involute-function-calculator
TG we-Calculate Editorial Team. "Involute Function Calculator — Gear Design." TG we-Calculate. 2026. https://we-calculate.com/calculator/involute-function-calculator.
TG we-Calculate Editorial Team, "Involute Function Calculator — Gear Design," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/involute-function-calculator
@misc{wecalculate_involute_function_calculator, title = {Involute Function Calculator — Gear Design}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/involute-function-calculator}}, year = {2026}, note = {TG we-Calculate} }
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