How to calculate gear ratio: single-stage, compound and planetary gear ratio formulas (with torque and speed)

What is a gear ratio?

The ratio i is input speed divided by output speed. Meshing gears share the same pitch-line velocity, so speed is inversely proportional to tooth count: one turn of the pinion turns the wheel z₁ / z₂ of a turn.

i = z₂ / z₁ = n₁ / n₂ = d₂ / d₁

A 15-tooth pinion driving a 45-tooth gear gives i = 3: a 3000 rpm motor becomes 1000 rpm. i > 1 reduces speed, i < 1 increases it. Two external gears turn in opposite directions; an idler reverses rotation again without changing the ratio.

TIPRatios are written 3:1, 1:3 or simply "ratio 3" depending on the industry. Always confirm which side is the input before comparing specs.

Speed, torque and power

n₂ = n₁ / i　　T₂ = T₁ · i · η　　P (kW) = T (N·m) × n (rpm) / 9549

Gears don't create energy: speed falls by i, torque rises by i, and efficiency η takes its share. A 0.5 N·m motor at 3000 rpm delivers about 157 W; through an i = 9 gearbox at 96% it gives 333 rpm and 4.32 N·m — still about 151 W.

These are ballpark figures. Ground steel gears reach 99%; printed PLA or PETG gears with rough flanks and backlash manage about 90–95% per stage; worm drives depend heavily on lead angle and lubrication.

Compound gear trains: multiply the stages

i total = (z₂/z₁) × (z₄/z₃) × (z₆/z₅) …　　η total = η₁ × η₂ × η₃ …

In a compound train the first-stage gear and second-stage pinion share a shaft. With 15→45 and 12→36: i = 3 × 3 = 9 and, at 98% per stage, total efficiency 0.98² ≈ 96%. Doing 9:1 in one stage would need a 135-tooth gear for a 15-tooth pinion.

• Keep each stage at ≤ 6–8; beyond that add stages or use a planetary or worm drive.
• Start each stage at the n-th root of the total ratio: i = 20 in two stages ≈ 4.47 × 4.47, in practice 4 × 5 (15→60, 15→75).
• A 20° pinion with fewer than 17 teeth undercuts; positive profile shift helps. For 3D printing stay at ≥ 12 teeth with shift.
• Backlash accumulates and is multiplied by the downstream ratios, so positioning gearboxes should use as few stages as possible.

TIPPick co-prime tooth counts (15:46 rather than 15:45) so every tooth meets every other tooth in turn. This "hunting tooth" ratio evens out wear — especially useful for printed gears.

Planetary gear ratio: why 1 + Zr/Zs?

A planetary set has a sun (Zs), several planets (Zp), a ring gear (Zr) and a carrier. The usual layout fixes the ring, drives the sun and takes output from the carrier. Because the carrier itself rotates, the ratio is not simply Zr/Zs — it follows from the Willis equation:

(ns − nc) / (nr − nc) = −Zr / Zs　→　ring fixed (nr = 0): i = ns / nc = 1 + Zr / Zs

Assembly: Zr = Zs + 2·Zp　　(Zs + Zr) / N = integer (N = number of planets)

Zs = 12, Zp = 24, Zr = 60 gives i = 1 + 60/12 = 6 and (12 + 60)/3 = 24, so three planets fit. Zs = Zp = 20, Zr = 60 gives i = 4 with four planets ((20 + 60)/4 = 20) but not three. Neighbouring planets must also clear each other: planet tip diameter < 2a·sin(180°/N), where a is the sun–planet centre distance.

TIPThe internal gear generator's "Planetary m1 · 60T" preset matches these tooth counts. Printed planetary gearboxes often use herringbone teeth, which self-centre and can't slide out axially.

Worked example: from requirement to tooth counts

1. Input: NEMA 17 stepper giving about 0.4 N·m at 600 rpm.
2. Requirement: 100 rpm and at least 2 N·m, so i = 600 / 100 = 6.
3. Torque check: T₂ = 0.4 × 6 × 0.95 ≈ 2.28 N·m — enough (apply a 1.5–2 safety factor for critical use).
4. Layout: one planetary stage Zs 12 / Zp 24 / Zr 60 gives exactly i = 6 on a coaxial output, or two spur stages 15→37 and 15→36 (i ≈ 5.92).
5. Centre distance: spur a = m(z₁ + z₂)/2 = 26 mm for m1, 15→37; sun–planet a = 1 × (12 + 24)/2 = 18 mm.
6. Check efficiency, speed and direction in the gear ratio calculator, then download the gears from the generators.