Gear Module Calculator

How Does Gear Module Define Gear Geometry?

Module (m) is the fundamental parameter of metric gear design, defined as the pitch diameter divided by the number of teeth. It is the inverse of diametral pitch used in imperial systems (DP = 25.4/m). Module determines the tooth size — larger module means larger, stronger teeth that transmit more torque but run less smoothly.

Standard involute spur gear proportions follow ISO 21771: addendum = 1.0m (tooth height above pitch circle), dedendum = 1.25m (below pitch circle), clearance = 0.25m. The pitch diameter (d = m × z) is the theoretical circle where two mating gears roll without slipping. The center distance between two gears = m × (z₁ + z₂) / 2.

Standard modules follow a geometric series per ISO 54: 0.5, 0.8, 1.0, 1.25, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 8.0, 10.0, 12.0, 16.0, 20.0. Using standard modules ensures availability of cutting tools (hobs, shaping cutters) and reduces manufacturing cost.

Formula: Pitch Diameter: d = m × z Addendum: ha = 1.0 × m Dedendum: hf = 1.25 × m Center Distance: a = m × (z₁ + z₂) / 2 Circular Pitch: p = π × m

Example Calculation

A gear pair with m = 2.5, z₁ = 20, z₂ = 45: Pitch diameter₁ = 2.5 × 20 = 50 mm. Pitch diameter₂ = 2.5 × 45 = 112.5 mm. Center distance = 2.5 × (20 + 45)/2 = 81.25 mm. Addendum = 2.5 mm, Dedendum = 3.125 mm. Circular pitch = π × 2.5 = 7.854 mm.

When to Use This Calculator

• Designing a gear pair and needing to determine pitch diameters and center distance from the selected module and tooth counts
• Reverse-engineering an existing gear — measuring the outside diameter and counting teeth to calculate the module
• Selecting a standard module from the ISO 54 series that provides adequate tooth strength for the transmitted torque
• Verifying gear mesh compatibility — two gears must have the same module and pressure angle to mesh correctly

Common Mistakes to Avoid

• Confusing module with diametral pitch — module (mm) = 25.4/DP (teeth per inch); using the wrong system produces gears that cannot mesh
• Designing with fewer than 17 teeth at 20° pressure angle without profile shift — this causes undercutting that weakens the tooth root and reduces contact ratio
• Measuring pitch diameter from the outside diameter without accounting for addendum — OD = pitch diameter + 2× module for standard gears
• Ignoring backlash requirements — zero backlash causes binding under thermal expansion; typically 0.03-0.05× module backlash is specified

How to Interpret Results

• If the module was derived from a measured pitch diameter, round to the nearest standard ISO 54 module (1.0, 1.25, 1.5, 2.0...) — a non-standard module means special tooling or a measurement error
• Center distance is the theoretical zero-backlash value — real assemblies add backlash (typically 0.03-0.05 × module) through center distance allowance or tooth thinning
• Addendum (1.0m) and dedendum (1.25m) combine to a whole depth of 2.25m — verify a physical gear with OD = pitch diameter + 2 × module before trusting the derived module
• The base circle diameter assumes a 20° pressure angle — the involute profile exists only outside this circle, and two gears mesh correctly only if both module and pressure angle match
• These results are geometry only — tooth bending and contact strength must be rated separately (AGMA 2001, ISO 6336) before committing a power transmission design

Related Standards & References

• ISO 21771 — Gears — Cylindrical involute gears and gear pairs — Concepts and geometry
• ISO 54 — Cylindrical gears for general engineering — Standard modules
• AGMA 913-B03 — Method for specifying the geometry of spur and helical gears
• AGMA 2001-D04 — Fundamental rating factors and calculation methods for involute spur and helical gear teeth
• DIN 3960 — Definitions, parameters and equations for involute cylindrical gears and gear pairs

Frequently Asked Questions

How do I select the right module for my application?

Module selection balances tooth strength, noise, and size. Calculate the required module from torque: m ≈ ∛(2T/(σ_b × y × b × z)) where T is torque, σ_b is allowable bending stress, y is Lewis form factor, b is face width, and z is tooth count. Then round up to the nearest standard module. For quiet operation (e.g., instruments), use smaller module with more teeth. For heavy loads (e.g., mining equipment), use larger module.

What is the minimum number of teeth to avoid undercutting?

For standard 20° pressure angle involute gears, the minimum number of teeth to avoid undercutting is 17. Below this, the dedendum interferes with the mating gear's addendum, weakening the tooth root. Solutions for fewer teeth: use profile shift (addendum modification x > 0), increase pressure angle to 25°, or use stub teeth (shorter addendum). In practice, 14-tooth minimum is common with positive profile shift.

How do I convert between module (metric) and diametral pitch (imperial)?

Module (mm) = 25.4 / Diametral Pitch (teeth/inch). For example, DP 10 = 25.4/10 = module 2.54, which would round to standard module 2.5. Common equivalences: DP 32 ≈ m0.8, DP 20 ≈ m1.25, DP 12 ≈ m2.0, DP 8 ≈ m3.0. When converting existing designs, always round to the nearest standard module per ISO 54.