Every dimension of a spur gear flows from one number: the module, which is simply the pitch diameter divided by the tooth count. Pick a module and a tooth count and the pitch circle is fixed (d = m·z); the teeth then stand one module proud of it and sit 1.25 modules below it, giving the outside and root diameters. Because two gears only mesh if they share the same module, it doubles as the compatibility code — and the gap between two meshing shafts, the centre distance, is just half the sum of their pitch diameters.
Reviewed: June 19, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: ISO 53 / standard full-depth gear-tooth proportions.
The gear-geometry equations
The addendum (height above the pitch circle) is one module and the dedendum (depth below) is 1.25 modules, so the tip diameter adds two addenda to the pitch diameter and the root subtracts two dedenda. Circular pitch — the arc distance from one tooth to the next along the pitch circle — is π·m, which is why module and π set the physical tooth spacing. Tooth thickness on the pitch circle is half of that, π·m/2.
Worked example — a module-2 gear pair
Scenario: A module-2 gear with 40 teeth meshing a 20-tooth pinion (20° pressure angle).
The 40-tooth gear has an 80 mm pitch circle, an 84 mm blank diameter to turn before cutting teeth, and a 75 mm root. Meshed with the 20-tooth pinion (40 mm pitch diameter) the shaft centres sit 60 mm apart, and the pair gives a 2:1 reduction. Keep both gears at module 2 and 20° pressure angle and they will run together correctly.
Frequently Asked Questions
The metric tooth size: m = pitch diameter ÷ teeth (mm). A module-2, 40-tooth gear has an 80 mm pitch circle. Gears must share module to mesh.
d = m × z. Module-2, 40 teeth → 80 mm. The pitch circle is the reference for all other gear dimensions.
Module is metric (mm/tooth); DP is imperial (teeth/inch). module = 25.4 ÷ DP. They are inversely related.
C = (d₁+d₂)/2 = m(z₁+z₂)/2. A module-2 20T + 40T pair sits 60 mm between centres.
Addendum = m (above pitch), dedendum = 1.25 m (below). Whole depth 2.25 m. Hence OD = m(z+2), root = m(z−2.5).