Planetary gear ratios, explained

A planetary set is the only common gear train where the same hardware gives three different ratios — you choose one by deciding which member sits still. Sun in the middle, planets around it on a rotating carrier, ring around everything: hold any one of the three and take the drive in and out through the other two.

One geometry constraint, three ratios

R = S + 2P · ring fixed: i = 1 + R/S · sun fixed: i = 1 + S/R · carrier fixed: i = −R/S

The tooth counts must close geometrically — the planet spans the gap between sun and ring, so P = (R − S)/2 must come out a whole number. The ratios are the Willis equation evaluated for each fixed member; notice the planets appear in none of them. The planetary gear calculator runs all of this — ratio, planet teeth and output speed — from the two counts that matter.

Worked example — 24-tooth sun, 72-tooth ring

One set, S = 24, R = 72 (so P = 24), input at 3,000 RPM:

ring fixed: 1 + 72/24 = 4:1 → 750 RPM · sun fixed: 1 + 24/72 = 1.33:1 → 2,250 RPM · carrier fixed: −72/24 = 3:1 reversed → −1,000 RPM

Same gears, three gearboxes: a 4:1 reduction, a gentle 1.33:1, and a reversing 3:1. That switchability is why automatics, hub gears and winch drives are built on planetaries — a brake band on the ring or a clutch on the carrier changes the ratio without moving a gear out of mesh. Stack two ring-fixed stages and ratios multiply: 4 × 4 = 16:1 in two compact stages.

What the planets do instead

Since they cancel out of the ratio, the planets' job is mechanical: three or four of them share the tooth load, which is why a planetary carries several times the torque of a same-size parallel-shaft pair. Two assembly checks make the set buildable: R − S even (whole planet teeth), and (S + R)/N a whole number for N equally spaced planets. The 24/72 set gives 96/3 = 32 — three planets drop straight in, and four (96/4 = 24) work too. Tooth geometry itself is ordinary involute work — see the involute gear calculator and the involute geometry guide — and simple two-shaft ratios live in the gear ratio calculator and the gear ratio guide.

Common mistakes

• Looking for planet teeth in the ratio. They aren't there — planets are idlers. If a catalog ratio doesn't match your tooth counts, check which member is fixed, not the planets.
• Dropping the sign. Carrier-fixed output reverses. In a multi-stage train a missed reversal flips the final rotation — the calculator keeps the ratio signed for exactly this reason.
• Tooth counts that can't be built. R − S odd means a fractional planet tooth; (S + R) not divisible by the planet count means the planets only assemble unequally spaced (possible, but a special layout). Check both before cutting anything.
• Forgetting torque scales with the ratio. A 4:1 ring-fixed stage quadruples output torque (less losses) — the carrier shaft, its bearings and the ring housing reaction all need to be sized for it, not for the input torque.

Frequently asked questions

How do you calculate a planetary gear ratio?

Only the sun and ring tooth counts matter. Ring fixed (sun in, carrier out): i = 1 + R/S. Sun fixed (ring in, carrier out): i = 1 + S/R. Carrier fixed (sun in, ring out): i = −R/S, the minus meaning the output reverses. A 24-tooth sun in a 72-tooth ring gives 4:1, 1.33:1 or a reversing 3:1 from the same hardware.

Do the planet teeth affect the ratio?

No. The planets are idlers — they must satisfy the geometry P = (R − S)/2 to fit between sun and ring, and they share the load, but they cancel out of the Willis equation. Changing planet count changes load capacity and assembly options, never the ratio.

How do you get reverse rotation from a planetary set?

Hold the carrier. With the planets unable to orbit they act as fixed idlers between sun and ring, and the output turns opposite the input at R/S — the 24/72 set delivers 3:1 reversed. This carrier-fixed arrangement is often called a star gear set.

What are the assembly conditions for a planetary gear set?

Two checks before any ratio is real: R − S must be a positive even number, because the planet needs a whole tooth count P = (R − S)/2; and for equally spaced planets, (S + R) divided by the planet count must be a whole number. The 24/72 set passes both — P = 24, and 96/3 = 32 allows three planets (96/4 = 24 allows four).