gear spur calculator - Aaron Graves, PhDude Replica

Spur Gear Calculator (Metric Module)

Use this tool to quickly compute core spur gear dimensions for one gear or a mating pair.

Pinion Teeth, z₁

Integer count of teeth on gear 1.

Gear Teeth, z₂ (optional)

If entered, ratio and center distance are calculated.

Module, m (mm)

Metric tooth size. Common values: 1, 1.5, 2, 2.5, 3, 4.

Pressure Angle, α (deg)

Standard involute gears often use 20°.

Addendum Coefficient, h_a*

Standard full-depth value: 1.0

Dedendum Coefficient, h_f*

Standard full-depth value: 1.25

What this gear spur calculator computes

This calculator provides the most common geometry values used in spur gear design, prototyping, CNC machining, and 3D printing. It is based on standard involute spur gear relationships in module units (millimeters). You can evaluate a single gear or a pair.

Pitch diameter
Outside (tip) diameter
Root diameter
Base diameter
Circular pitch and tooth thickness at pitch circle
Whole depth, addendum, dedendum, and clearance
Gear ratio and center distance for a mating pair

Input definitions

Teeth count (z)

The number of teeth strongly affects ratio, diameter, and the chance of undercut in small pinions. For many 20° full-depth gears, values below roughly 17 teeth may require profile shift to avoid undercut.

Module (m)

Module controls overall gear size. Larger module means larger teeth and larger diameters for the same teeth count. Pitch diameter is directly proportional to module.

Pressure angle (α)

Pressure angle influences contact geometry and base circle size. A 20° pressure angle is the modern default for general-purpose gears.

Core formulas used

Pitch diameter: d = m × z
Outside diameter: d_a = d + 2h_a = m(z + 2h_a*)
Root diameter: d_f = d − 2h_f = m(z − 2h_f*)
Base diameter: d_b = d × cos(α)
Circular pitch: p = πm
Tooth thickness at pitch line: s = p/2
Center distance (pair): a = (d₁ + d₂)/2 = m(z₁ + z₂)/2
Gear ratio (pair): i = z₂ / z₁

Practical design tips

Keep both gears at the same module and pressure angle.
Check root diameter remains positive and manufacturable.
For very small pinions, consider profile shift or helical alternatives.
Account for backlash and tolerance in printed or machined parts.
Validate load capacity and material strength separately from geometry.

Example

If z₁ = 20, z₂ = 40, m = 2.5, α = 20°: pitch diameters are 50 mm and 100 mm, center distance is 75 mm, and gear ratio is 2:1. That means the larger gear rotates at half the speed of the pinion.

Important note

This tool covers geometric sizing only. For a production-ready transmission, include torque, bending stress, contact stress, lubrication, material hardness, quality class, and safety factors in your full engineering workflow.