Free gear calculator Gear calculator

Planetary gear ratio calculator

A planetary gear ratio calculator gives the speed ratio of a sun, planet and ring gear train for any choice of fixed, input and output member, and checks that the tooth counts can be assembled.

Enter the tooth counts, pick the member that is held, and read the ratio, direction, output speed and torque. The drawing updates with every change and the link keeps your numbers.

Ratio, input to output

5 : 1

Reduction. The output turns the same way as the input, 5 times slower.

Exact ratio: 5/1
Output speed: Enter an input speed
Output torque, ideal: Enter an input torque

Pass: Ring = sun + 2 × planet: 18 + 2 × 27 = 72.
Pass: Equal spacing: (18 + 72) / 3 = 30, a whole number.
Pass: Neighboring planets clear: (18 + 27) × sin(180° / 3) = 38.97 > 27 + 2 = 29. Tip circles only; running clearance is not included.
Pass: Sun and planets have 17 or more teeth, so standard 20° teeth do not undercut.

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Need this stage built as a prototype? See custom planetary gearboxes for robot joints.

Input Output Held

How do you calculate a planetary gear ratio?

Planetary gear trains are also called epicyclic gear trains. Every simple planetary train (one set of planets meshing with both the sun and the ring) obeys one relation between the speeds of the sun (ωs), the ring (ωr) and the carrier (ωc), known as the Willis equation:

(ωs − ωc) / (ωr − ωc) = −Zr / Zs

Zs and Zr are the sun and ring tooth counts. The planet count does not change the ratio.

Hold one member (its speed is zero), drive a second and take the output from the third. The ratio i is input speed divided by output speed. A negative ratio means the output turns opposite to the input.

The six configurations, with the 18/27/72 example
Held	Input	Output	Ratio i	18/27/72	Result
Ring	Sun	Carrier	`1 + Zr / Zs`	`1 + 72 / 18 = 5`	Reduction, same direction
Ring	Carrier	Sun	`Zs / (Zs + Zr)`	`18 / 90 = 0.2`	Speed increase, same direction
Sun	Ring	Carrier	`1 + Zs / Zr`	`1 + 18 / 72 = 1.25`	Reduction, same direction
Sun	Carrier	Ring	`Zr / (Zs + Zr)`	`72 / 90 = 0.8`	Speed increase, same direction
Carrier	Sun	Ring	`−Zr / Zs`	`−72 / 18 = −4`	Reduction, opposite direction
Carrier	Ring	Sun	`−Zs / Zr`	`−18 / 72 = −0.25`	Speed increase, opposite direction

Which tooth counts can actually be assembled?

A ratio is only useful if the gears fit together. For standard gears of one module, three conditions apply:

Concentric: the planets mesh with the sun and the ring at the same center distance, so Zr = Zs + 2 × Zp.
Equal spacing: N planets can be spaced evenly only when (Zs + Zr) / N is a whole number.
Neighbor clearance: the tip circles of neighboring planets must not touch, so for standard full-depth teeth (Zs + Zp) × sin(180° / N) > Zp + 2.

Profile-shifted gears change the first condition. The calculator flags those tooth counts as outside its scope; they are worked out in the design review. It also warns when the sun or planets have fewer than 17 teeth, the point below which standard 20° teeth undercut.

How do output speed and torque change?

output speed = input speed / i

ideal output torque = input torque × i

With an efficiency η entered, the calculator multiplies the ideal torque by η once, as the overall efficiency of all stages.

Real efficiency depends on the design, lubrication, speed and load, and it is measured on a built gearbox. The calculator never assumes a value: it uses one only if you enter it.

Worked example: the 18/27/72 planetary stage

18 + 2 × 27 = 72, so the stage is concentric.
(18 + 72) / 3 = 30, a whole number, so three planets can be equally spaced.
(18 + 27) × sin(60°) ≈ 38.97, which is more than 27 + 2 = 29, so the planets clear each other.
Ring held, sun in, carrier out: 1 + 72 / 18 = 5, a 5:1 reduction. 3,000 rpm in gives 600 rpm out.

This stage is the design study drawn across this site, including the exploded view on the Pico home page. It illustrates the math and is not a catalog product.

How do multi-stage planetary gearboxes multiply the ratio?

Stages in series multiply: two 5:1 stages give 25:1, three give 125:1. Each added stage also adds meshes, bearings and clearances, so losses and the sources of backlash grow with the stage count. For a robot joint, that trade is part of choosing a reducer type, covered on the page about robot joint gearboxes and reducers.

Once the numbers work, the next step is a drawing and a build. See custom gears and gear sets for robotics or start a project inquiry.

Questions

What is the formula for a planetary gear ratio?

With the ring gear fixed, the sun as input and the carrier as output, the ratio is 1 + Zr/Zs, where Zr and Zs are the ring and sun tooth counts. For example, 18 sun teeth and 72 ring teeth give 1 + 72/18 = 5, a 5:1 reduction.

Why does the calculator reject some tooth counts?

For standard gears of one module, the ring must have Zr = Zs + 2Zp teeth. For N equally spaced planets, (Zs + Zr)/N must be a whole number, and neighboring planets must not touch: (Zs + Zp) × sin(180°/N) must be greater than Zp + 2.

Does the calculator include efficiency?

Only if you enter it. The calculator gives ideal torque from the ratio; enter an efficiency estimate to see its effect. Real efficiency depends on the design, lubrication and load, and is measured on a built gearbox.

Can I calculate a two-stage planetary gearbox?

Yes, for identical stages: set the number of stages and the calculator multiplies the stage ratios, so two 5:1 stages give 25:1. For stages with different tooth counts, calculate each stage and multiply the ratios.

Is the 18/27/72 gearbox a Pico product?

No. It is a design study used to illustrate the math. If you need a planetary stage like it built as a prototype, describe it in a project inquiry.

Need this stage built?

Send the tooth counts, the loads and the quantity for a first build. Pico reviews the design and replies with a proposed scope for a prototype or pilot batch.

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