Spur Gear Pair Designer

Pitch, outside, root and base diameters, center distance, contact ratio and Lewis tooth stress for a spur gear pair, with a to-scale involute drawing.

How it works

For standard full-depth involute spur gears (no profile shift) every dimension follows from the module m, the numbers of teeth z and the pressure angle α.

d = m z, da = m (z + 2), df = m (z − 2.5), db = d cos α, C = (d₁ + d₂) ÷ 2

mc = [√(ra1² − rb1²) + √(ra2² − rb2²) − C sin α] ÷ (π m cos α)

σ = Wt Kv ÷ (b m Y)

The Lewis equation is the simplest strength check; full design uses AGMA or ISO 6336 with load, dynamic, size and reliability factors, and a surface-durability (pitting) check. The principles are on Gear Design.

Worked example

A 3 mm module pair with 20 and 50 teeth at a 20° pressure angle. The pitch diameters are 60 mm and 150 mm, so the center distance is 105 mm and the ratio is 2.5. The outside diameters are 66 mm and 156 mm. The contact ratio is 1.656.

Transmitting 2 kW at 1 450 rpm: the pinion torque is 13.17 N·m, the tangential force is 439.0 N at a pitch-line speed of 4.56 m/s, so Kv = 2.494. With a 25 mm face and Y = 0.322, the Lewis stress is 45.3 MPa, comfortably below the bending endurance of hardened steel gears.

Things to know

With 20° teeth, fewer than 18 teeth are undercut unless profile shifted (the limit is 2 ÷ sin²α = 17.1). A 14.5° tooth needs 32.

Two gears only mesh if they have the same module and pressure angle. Backlash is not drawn: the drawing shows zero backlash.

The Lewis stress treats the tooth as a cantilever with the load at the tip. It is conservative about load sharing but ignores stress concentration; use a safety factor of 1.5 or more.

Surface fatigue (pitting) often governs hardened gears: check contact stress as well.

Helical, bevel and worm gears need different equations; the ratio and speeds follow the same rule.

Common questions

What is the difference between module and diametral pitch?

Module is pitch diameter in millimeters per tooth (m = d ÷ z). Diametral pitch is teeth per inch of pitch diameter (DP = z ÷ d). They are linked by m = 25.4 ÷ DP.

Why does the contact ratio matter?

A contact ratio of 1.7 means that for 70 percent of the time two pairs of teeth share the load and for the rest one pair carries it. Higher values are smoother and quieter.

How many teeth should the pinion have?

At least 17 for 20 degree teeth without profile shift, and more (20 to 30) for good strength, smooth action and low sliding.

Sources

  • Shigley’s Mechanical Engineering Design, Chapters 13 and 14 (gear geometry, Lewis equation).
  • ISO 53 and ISO 54 (standard basic rack and modules); ISO 6336 and AGMA 2001 (rating).

Last reviewed: November 2, 2026