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MachineCalcs

Involute Gear Calculator

Tooth geometry from module (or diametral pitch), pressure angle and tooth count — with a live tooth-profile render and DXF export. Metric and imperial.

Inputs

mm

Results

Pitch diameter(d)
40 mm

Module (mm)
2

Diametral pitch (1/in)
12.7

Base diameter(d_b)
37.59 mm

Outside (tip) diameter(d_a)
44 mm

Root diameter(d_f)
35 mm

Addendum
2 mm

Dedendum
2.5 mm

Whole depth
4.5 mm

Circular pitch(p)
6.283 mm

Tooth thickness (at pitch)(s)
3.142 mm

Center distance (with mate)(a) ✓
60 mm

Recommended backlash
0.08 mm

Typical 0.03–0.05 × module.

  • Standard full-depth involute teeth (addendum = m, dedendum = 1.25 m, clearance = 0.25 m).
  • Minimum teeth to avoid undercut at 20° is 18 without profile shift.

How it works

An involute spur gear is defined by its module m (or diametral pitch), pressure angle α and tooth count z. The pitch diameter is d = m · z and the tooth flanks are involutes of the base circle d_b = d·cos α. Standard full-depth teeth use addendum a = m, dedendum b = 1.25 m and clearance 0.25 m, giving a tip diameter d_a = d + 2m and root diameter d_f = d − 2.5m. The tooth thickness at the pitch circle is s = m·(π/2 + 2x·tan α), where x is the profile-shift coefficient.

With a mating gear, the standard center distance is a = m·(z + z₂)/2. When a profile shift is applied the gears run on a larger working pressure angle α_w, found by inverting the involute function inv α = tan α − α; the calculator solves it numerically and adjusts the center distance accordingly. It also flags undercut — at 20° a standard tooth is undercut below about 17 teeth unless you add a positive shift.

Worked example

A module-2 mm, 20° pressure-angle pinion with z = 20 teeth meshing with a z₂ = 40 gear, no profile shift. The pitch diameter is d = 40 mm, base diameter 37.59 mm, tip diameter 44 mm and root diameter 35 mm. The tooth thickness at the pitch circle is π ≈ 3.14 mm, the circular pitch is 6.28 mm, and the center distance is m·(z+z₂)/2 = 60 mm. With 20 teeth there is no undercut (the 20° minimum is ≈ 17). The render and DXF show this exact profile.

Reference data

Preferred metric modules (ISO 54, series 1), with the circular pitch and the approximate equivalent diametral pitch. Sticking to a preferred module makes tooling, mating gears and replacements easier to source.

ISO 54 series-1 preferred modules. Circular pitch p = π·m; equivalent DP = 25.4 / m.
Module (mm) Circular pitch (mm) ≈ Diametral pitch (1/in) Notes
1 3.142 25.4 Fine-pitch instruments and small mechanisms.
1.25 3.927 20.3
1.5 4.712 16.9
2 6.283 12.7 Common general-machinery module.
2.5 7.854 10.2
3 9.425 8.47
4 12.57 6.35
5 15.71 5.08
6 18.85 4.23
8 25.13 3.18
10 31.42 2.54 Heavy power transmission.
12 37.7 2.12
16 50.27 1.59
20 62.83 1.27

Source: ISO 54 preferred-module series. Diametral pitch shown for reference; DP gears use their own standard series.

Frequently asked questions

How do I calculate involute gear tooth geometry?
From the module m and tooth count z: pitch diameter d = m·z, addendum = m, dedendum = 1.25 m, tip diameter = d + 2m and root diameter = d − 2.5 m. Enter the module (or diametral pitch), pressure angle and tooth count above and the calculator returns the full geometry.
What is the difference between module and diametral pitch?
Module (mm) is the pitch diameter divided by the number of teeth; diametral pitch (1/in) is the number of teeth divided by the pitch diameter in inches. They are reciprocals scaled by 25.4: m = 25.4 / DP. Use the size-system selector to enter whichever your drawing uses.
Which pressure angle should I use?
20° is the modern standard and the right default. 14.5° is legacy (older equipment); 25° gives a stronger tooth but more separating force and noise. Mating gears must share the same module and pressure angle.
How many teeth can I have before undercut?
For a standard 20° tooth, undercut begins below about 17 teeth. Use more teeth, or add a positive profile shift. The calculator warns when undercut is likely and suggests the minimum shift coefficient.
What does the DXF export give me?
It downloads the exact involute tooth profile of the gear as a closed polyline (LWPOLYLINE), ready to open in CAD/CAM for laser, waterjet or wire-EDM cutting — the part that gear-generator searches actually want.
Does this work in metric and imperial?
Yes — toggle SI/Imperial in the header to switch diameters between mm and inches. The module-versus-diametral-pitch choice is separate, set by the size system.

Method & assumptions

  • Standard full-depth involute profile (addendum = m, dedendum = 1.25 m). Stub or non-standard tooth systems differ.
  • The profile-shifted center distance assumes the mating gear is unshifted; for a shifted pair, use the sum x₁ + x₂.
  • Recommended backlash is a module-based rule of thumb (≈ 0.03–0.05 × module); real backlash is set by tooth-thickness allowance and the actual center distance and its tolerance.
  • The DXF profile is a geometric involute with a simplified root fillet — verify against your manufacturing requirements.

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