How to use this calculator
- Enter the sun and ring teeth. Enter the sun (S) and ring (R) tooth counts. The ring must exceed the sun and R − S must be even.
- Choose the fixed member. Select which member is held stationary — ring, sun or carrier. This sets the ratio via the Willis equation.
- Enter the input speed. Enter the speed of the driving member.
- Read the results. Read the gear ratio, the planet tooth count (R − S)/2, and the output speed.
How it works
A simple planetary (epicyclic) set has three members — the central
sun, the outer ring (annulus), and the
carrier that holds the planets. Holding one member fixed
sets the ratio (the Willis equation): with the
ring fixed it is i = 1 + R/S,
with the sun fixed it is i = 1 + S/R, and with
the carrier fixed it is i = −R/S — the minus
sign reverses the output direction. Output speed is n₂ = n₁ / i.
The ratio depends only on the sun and ring tooth counts, never on the planet
count. The planets must satisfy R = S + 2P, so the planet teeth
are P = (R − S)/2 — which requires R − S to be a positive even
number.
For fixed-axis pairs, use the gear ratio calculator or the gear tooth ratio calculator instead. Once the epicyclic ratio is chosen, the mating gears still need standard tooth geometry: check module, center distance and undercut with the involute gear calculator, then estimate tooth loads with the gear mesh force calculator.
Worked example
Verified against the live calculator
A 24-tooth sun and a 72-tooth ring with the ring fixed give
i = 1 + 72/24 = 4 — a 4:1 reduction in the same
direction, with 24-tooth planets ((72 − 24)/2). At 3,000 RPM into
the sun, the carrier turns 750 RPM. Those are the numbers the
calculator shows for these inputs.
Frequently asked questions
How do I calculate a planetary gear ratio?
Pick which member is held fixed, then use the Willis equation on the sun (S) and ring (R) tooth counts. With the ring fixed and the sun driving the carrier, the ratio is 1 + R/S. With the sun fixed (ring drives the carrier) it is 1 + S/R. With the carrier fixed (sun drives the ring) it is −R/S, which also reverses the direction.
What is the gear ratio of a planetary set with the ring fixed?
With the ring (annulus) held stationary and the sun as input driving the carrier as output, the ratio is 1 + R/S. For a 24-tooth sun and a 72-tooth ring that is 1 + 72/24 = 4, a 4:1 reduction in the same direction.
How does the number of planet gears affect the ratio?
It doesn't. The ratio depends only on the sun and ring tooth counts. The planet teeth P just satisfy the geometry R = S + 2P, and the number of planets only affects load sharing and assembly, not the ratio.
How do I get a reverse-direction output from a planetary set?
Hold the carrier fixed and drive the sun, taking the output off the ring (or vice versa). The ratio is −R/S — the minus sign means the output turns opposite to the input. With the carrier free (ring or sun fixed), the output keeps the same direction.
Why does R = S + 2P for a planetary gear set?
The planets sit between the sun and the ring, so the ring radius equals the sun radius plus two planet radii. With the same module (tooth size), radius is proportional to tooth count, so R = S + 2P. That makes the planet count P = (R − S)/2 — which means R − S must be a positive even number.
Does this work in metric and imperial?
The gear ratio and planet count are pure tooth-count numbers, so they are unitless and identical in either system. Only the input and output speeds carry a unit (RPM, the same in SI and imperial); toggle SI/Imperial in the header.
Method & assumptions
- Single-stage simple planetary set with standard (same-module) meshing; one sun, one ring, equal planets.
- Ideal kinematics — the ratio ignores efficiency; real planetary stages lose a few percent per mesh.
- Geometry must allow assembly: equally-spaced planets require (R + S) / (number of planets) to be an integer, and R − S must be a positive even number. Tooth strength and bearing loads are separate checks.