Gear Module & Pitch
Calculate gear module, tooth pitch, pitch circle diameter, addendum/dedendum circles and centre distance for standard spur gears
Inputs
Formula Interpretation
Tooth Pitch
Tooth pitch = arc length on the pitch circle per tooth = × π / . Equals π × module for any gear with the same module.
Module
Module is the ratio of pitch circle diameter to tooth count . It is the fundamental size parameter — meshing gears must have the same module.
Pitch Circle Diameter
Pitch circle diameter = module × teeth . The pitch circle is the reference circle where tooth proportions are defined.
Addendum Circle
The addendum circle defines the tooth tips. Addendum height , so da = D + 2m = m(Z + 2).
Dedendum Circle
The dedendum circle defines the tooth roots. Dedendum (includes clearance c* = 0.25m), so df = D − 2.5m.
Full Tooth Depth
Full tooth depth = addendum + dedendum = m + 1.25m = 2.25m. Tooth height is approximately 2× the module.
Centre Distance
For a meshing pair, both gears share the same module. Centre distance = sum of pitch circle radii = ( + ) / 2.
Knowledge Points
Module as the Size Standard
The module m (in mm) is the fundamental gear-size parameter prescribed by national standards (ISO 54 series). All tooth proportions scale with m: pitch = πm, height ≈ 2.25m, fillet radii ≈ 0.38m. Only gears with the same module can mesh.
Pitch Circle and Pitch Point
The pitch circle is a virtual reference circle of diameter D = mZ; it is where the involute tooth profile is generated. For a meshing pair, the two pitch circles roll on each other without slipping, and they meet at the pitch point — the instantaneous contact line on the line of centres.
Standard Module Values
ISO 54 standard modules (mm): 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40, 50. Designers select the smallest standard module that satisfies the stress and deflection requirements, then finalise dimensions.
Worked Example
A gear with teeth meshes with a gear whose pitch circle diameter is and tooth count is . Find the pitch circle diameter of the first gear and the centre distance.
Step 1 — Find the module from the known gear
Step 2 — Pitch circle diameter of gear 1 (same module)
Step 3 — Centre distance of the meshing pair
Result: D₁ = 120 mm, C = 90 mm. Because both gears share the same module (m = 3), they will mesh correctly at this centre distance.
Extended Knowledge
- •In the imperial system, diametral pitch (unit: teeth/inch) is used instead of the module. The relationship between them is = 25.4/m. For example, a module of 3 mm corresponds to a diametral pitch of approximately 8.47 teeth/inch.
- •Helical gears use the normal module (perpendicular to the tooth direction) and the transverse module = /cos(β). The two differ by the helix angle β, and the center distance formula must be adjusted accordingly.
- •Profile shifted gears adjust the center distance by introducing a profile shift coefficient x without changing the module or the number of teeth. A positive shift (x>0) can increase the tooth root thickness and improve load-carrying capacity, while a negative shift thins the teeth. The shift amount is expressed as a multiple of the module.
Related Standards & Articles
Authoritative references for the formulas used in this calculator
Gear Design & Selection Guide: From Parameters to System Design
A practical guide to gear selection and system design — covering gear types, key parameters (module, tooth count, pressure angle), material choices, centre distance, backlash, and strength verification. Includes hard-won tips on tooth pairing, lubrication, and weight reduction design.
Gear Design Symbol Reference (GB/T 2821 / ISO 701)
A complete notation reference for gear design calculations — geometric parameters, load factors, strength properties, and safety factors — per GB/T 2821-2003 and ISO 701:1998, with full LaTeX symbol rendering.
Gear Transmission Types: Selection Overview (Involute, Bevel, Worm & Planetary)
A comprehensive comparison of all major gear transmission types — involute cylindrical, bevel, worm, planetary, few-tooth-difference, and harmonic drives — covering key characteristics, transmission ratios, power capacities, speed ranges, and typical industrial applications for mechanical design selection.