Spur Gear Calculator

Pick a tooth size, set the tooth counts and speed. Get the ratio, torque, every diameter and the center distance, with profile shift and a tooth profile you can copy into CAD.

Tooth size

module

Driver teeth

Driven teeth

Pressure angle

Profile shift x₁

Profile shift x₂

Gear type · helix angle

° β

mm b

0° is a spur gear. Helical gears run quieter but push sideways on the bearings.

Input speed

RPM

Input torque

N·m

Torque is optional. Output torque assumes about 98% efficiency for one spur gear pair.

Drag a rim dot to change the tooth count

Gear dimensions

Dimension	Driver	Driven

Gear pair finder

Enter the output speed you need. Pairs are sorted by speed error, then by size.

Target output speed

RPM

Driver gear

Driver	Driven	Ratio	Speed	Error

Tooth profile for CAD

Exact involute outline in millimetres, with a bore. Paste the SVG into Inkscape or a laser cutter app, or save the DXF text as a .dxf file for CAD and CNC.

Driver bore

Driven bore

Gear Train – Multi-Stage Reduction

Stack several gear pairs on shared shafts, like a gearbox. The driven gear of each stage sits on the same shaft as the driver of the next. Starts from the input speed and torque above.

Number of stages

Driver teethDriven teethModuleOutput

How it is calculated

Involute spur gears with full-depth teeth (addendum 1·m, dedendum 1.25·m). m is the module, z the tooth count, α the pressure angle, x the profile shift coefficient. For inch gears m = 25.4 / DP.

Gear ratio i = z₂ / z₁ Output speed n₂ = n₁ / i Output torque T₂ ≈ T₁ · i · 0.98 Pitch diameter d = m · z Base diameter db = d · cos α Working angle inv αw = inv α + 2·tan α·(x₁ + x₂)/(z₁ + z₂), inv θ = tan θ − θ Center distance aw = m·(z₁ + z₂)/2 · cos α / cos αw Tip shortening k = (x₁ + x₂) − (aw − a)/m Outside diameter da = d + 2·m·(1 + x − k) Root diameter df = d − 2·m·(1.25 − x) Tooth thickness s = m·(π/2 + 2·x·tan α) Contact ratio ε = (√(ra₁² − rb₁²) + √(ra₂² − rb₂²) − aw·sin αw) / (π·m·cos α) Undercut limit x_min = (z_min − z)/z_min, z_min = 2 / sin² α

With no shift, gears below about 17 teeth at 20° (32 at 14.5°) are undercut by a hob or rack cutter. A positive shift on the small gear fixes this and makes the tooth root stronger. Keep the tooth tip at least 0.25·m wide. A contact ratio above 1.2 gives smooth running; 1.4 to 1.8 is typical.