Gear Ratio Calculator for Speed, Torque and Gear Trains
Calculate a clearly defined reduction factor, signed speed multiplier, output RPM, rotation direction and torque estimate for a gear pair or multi-stage train. A separate simple-planetary mode solves sun, ring and carrier speeds without applying a fixed-axis formula to a moving carrier.
- External and internal pairs
- Up to 8 train stages
- Speed and torque estimates
- Simple planetary mode
Online Gear Ratio Calculator
Choose a fixed-axis gear pair, a serial gear train or a simple planetary set. Tooth ratios stay exact, unit conversions remain explicit, and each result states the direction and assumptions used.
A 20-tooth external driver turns a 60-tooth driven gear at one-third speed.
Detailed Results
| Quantity | Relationship | Result |
|---|---|---|
| Reduction factor | 60 / 20 | 3 |
| Output speed | 1200 / 3 | 400 rpm |
| Rotation | One external mesh | Counterclockwise |
Calculation Steps
- Divide 60 driven teeth by 20 driver teeth to obtain i = 3.
- Use output speed = input speed / i = 1200 / 3 = 400 rpm.
- Reverse direction because the pair has one external mesh.
- Multiply 10 N·m by 3 for 30 N·m ideal torque, then by 95% for 28.5 N·m.
This is a kinematic ratio and steady transmitted-torque estimate. It does not rate the teeth, shafts, bearings, lubrication or guard.
How to Use This Gear Ratio Calculator
- Choose the gear system. Select one pair, a multi-stage train or a simple planetary gearset.
- Enter tooth counts. Use positive whole numbers and identify every external or internal mesh correctly.
- Add operating data. Enter speed, direction, torque and efficiency when you want more than the tooth ratio.
- Set optional geometry. For a pair, choose module or diametral pitch only for compatible standard spur gears.
- Choose output units. Select speed, torque and displayed precision without changing the exact internal ratio.
- Calculate and review. Check the ratio convention, output direction, formula steps, geometry assumptions and safety limits.
What Does Gear Ratio Mean?
Gear-ratio conventions vary, so this page labels its convention instead of displaying an unexplained number. For fixed-axis gears, the reduction factor i equals driven teeth divided by driver teeth. Its magnitude also equals input speed divided by output speed in the ideal rigid, no-slip model.
An i value greater than 1 is a speed reduction. The output turns more slowly and the ideal torque magnitude rises by the same factor. An i value below 1 is an overdrive or speed increase. A value of 1 preserves speed magnitude. At a zero-speed state, the tooth-count ratio remains defined even though the instantaneous speed quotient is 0/0. This calculator also shows the signed speed multiplier k = noutput/ninput, because its sign carries direction information that a positive reduction factor cannot.
For two external spur gears, k = −Zdriver/Zdriven, so parallel shafts rotate in opposite directions when viewed from one common end. An external pinion meshing inside a ring gear gives a positive multiplier and the shafts rotate in the same direction.
Gear Pair Speed and Torque Formulas
| Quantity | Formula | Scope |
|---|---|---|
| Reduction factor | i = Z2/Z1 | Positive tooth-count ratio |
| External speed | n2 = −n1Z1/Z2 | Parallel external gears |
| Internal speed | n2 = +n1Z1/Z2 | Pinion driving an internal ring |
| Ideal torque magnitude | T2,ideal = T1i | Ideal steady power transfer |
| Estimated torque magnitude | T2 = ηT1i | Forward moving operation at assumed efficiency |
Efficiency changes the power and torque estimate, not the rigid tooth-speed relationship. Output speed remains set by tooth counts. At zero input speed, the output is stationary and no clockwise or counterclockwise direction exists. The calculator then withholds efficiency-adjusted torque because power-ratio efficiency becomes 0/0 at standstill, while still showing the ideal static torque multiplier for reference.
Worked Gear Ratio Examples
Example 1: 20-tooth driver and 60-tooth driven gear
The reduction factor is 60/20 = 3. At 1,200 rpm input, output speed is 1,200/3 = 400 rpm. An external mesh reverses direction. With 10 N·m input torque and 95% assumed efficiency, ideal output torque is 30 N·m and the estimated moving output torque is 28.5 N·m.
Example 2: speed-increasing pair
A 60-tooth driver turning a 20-tooth driven gear has i = 20/60 = 1/3. A 300 rpm input produces 900 rpm output. This is an overdrive. With 6 N·m input torque, ideal output torque is 2 N·m before losses.
Example 3: two-stage compound train
Stage one uses 12 driving teeth and 36 driven teeth, giving 3:1. Stage two uses 15 and 45 teeth, also giving 3:1. Multiply the stage factors: itotal = 3 × 3 = 9. At 1,800 rpm, output speed is 200 rpm. Two external meshes restore the original direction. Efficiencies of 90% and 95% multiply to 85.5%.
Example 4: ring-fixed planetary set
A simple set has 20 sun teeth and 60 ring teeth, so the standard planet gear has (60 − 20)/2 = 20 teeth. With the ring fixed and sun at 1,800 rpm, the carrier turns at 450 rpm in the sun direction. The reduction factor from sun input to carrier output is 1 + 60/20 = 4.
Simple, Idler and Compound Gear Trains
For serial fixed-axis stages, multiply every driven-to-driver tooth ratio. Multiply a negative sign for each external mesh and a positive sign for each internal mesh. Gears rigidly mounted to the same shaft share angular speed and direction.
A simple idler changes direction but does not change the endpoint magnitude ratio because its tooth count appears once in a numerator and once in a denominator. A compound intermediate shaft carries two different gears. Those distinct tooth counts do not cancel, so each stage factor matters. Enter consecutive meshes in their power-flow order. The calculator supports up to eight stages and lists each stage ratio, cumulative ratio, mesh sign and cumulative efficiency.
Module, Diametral Pitch and Centre Distance
The optional geometry result is limited to standard, unshifted spur reference geometry. With module m in millimetres per tooth, pitch diameter d = mZ. With diametral pitch Pd in teeth per inch, pitch diameter d = Z/Pd inches.
| Geometry | Module formula | Diametral-pitch formula |
|---|---|---|
| Driver or pinion diameter | d1 = mZ1 | d1 = Z1/Pd |
| External centre distance | a = m(Z1 + Z2)/2 | a = (Z1 + Z2)/(2Pd) |
| Internal centre distance | a = m(Zring − Zpinion)/2 | a = (Zring − Zpinion)/(2Pd) |
Matching module or diametral pitch alone does not prove that gears mesh safely. Pressure angle, helix geometry, tooth form, profile shift, backlash, interference and manufacturing tolerances still matter. For a helical pair, the relevant transverse geometry must be used. Profile-shifted gears may require a working centre distance different from the standard value shown here.
Simple Planetary Gear Ratio and Speed
A planetary set cannot use the ordinary driven-teeth/driver-teeth equation when its carrier moves. For a simple sun, internal ring and carrier, the signed angular speeds obey the Willis relation:
The calculator offers a one-fixed-member mode and a two-known-speed general mode. In fixed mode it chooses the remaining member as output and reports a defined input/output reduction. General mode solves any one speed from the other two, then identifies a fixed-member case when one speed is zero or a locked common-speed state when all three speeds match. Equal nonzero member speeds are 1:1 in absolute speed even though their carrier-relative Willis quotient is 0/0. In an all-zero stationary state, the instantaneous speed quotient is also 0/0. A remaining two-input differential operating point has no single selected input/output ratio.
For equal-module standard coaxial geometry, ring teeth must satisfy Zr = Zs + 2Zp. The calculator therefore requires the ring to be larger than the sun and their difference to be even. Planet count, equal-spacing assembly, planet-to-planet clearance, interference and load sharing are not checked. Planetary torque is also withheld because speed kinematics alone do not determine three-member torque distribution or the fixed-member reaction.
Rotational Speed and Torque Units
Tooth ratio is dimensionless. Rotational speed is converted through revolutions per second. One revolution per second equals 60 rpm, 2π radians per second and 360 degrees per second. Radians are dimensionless in SI, while rad/s remains a useful angular-speed expression.
Torque is normalized to newton-metres. One pound-force foot equals 1.3558179483314004 N·m, one pound-force inch equals 0.1129848290276167 N·m, and one kilogram-force metre equals 9.80665 N·m under standard gravity. Torque is not energy even though both N·m and joule reduce to related SI base dimensions. Rotational work depends on torque multiplied by angular displacement.
Accuracy, Efficiency and Design Limits
- Use integer teeth. The calculator rejects fractional, zero and negative tooth counts.
- Declare the mesh. An internal mesh has a different direction sign and requires a larger driven ring in pair and stage modes.
- Treat efficiency as an assumption. It varies with load, speed, lubrication, temperature, alignment and backdriving direction.
- Do not infer capacity. A torque result is not a tooth-bending, contact-stress, shaft or bearing rating.
- Check transient loads. Shock, starting inertia, reversal, backlash and resonance may exceed steady estimates.
- Keep precision realistic. Exact tooth ratios do not remove uncertainty in speed, torque or efficiency measurements.
The tool models rigid kinematics and a simple forward power-efficiency estimate. It excludes worm, bevel, crossed-axis, rack-and-pinion and continuously variable systems. It also does not select pressure angle, module, face width, material, heat treatment, lubrication or service factor. Use the Power Calculator to review the separate relationship P = Tω.
Machine Safety and Guarding
Exposed gears create in-running nip, entanglement, crushing and ejection hazards. A numerical ratio does not establish a safe machine. Guard power-transmission components, control stored energy and prevent unexpected startup under the requirements and procedures that apply to the equipment and workplace.
Do not reach into moving gearing to confirm direction, backlash or speed. De-energize, isolate and verify the machine before inspection or adjustment. Consequential reducer selection and machine design require qualified review, current standards, manufacturer data and documented load cases.
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Frequently Asked Questions
How do I calculate gear ratio from teeth?
Using this page's convention, divide driven teeth by driver teeth: i = Zdriven/Zdriver. A 60-tooth driven gear and 20-tooth driver give i = 3, or a 3:1 reduction.
How do I calculate output RPM from gear ratio?
Divide input RPM by the positive reduction factor. A 1,200 rpm input with i = 3 produces 400 rpm output. Direction is handled separately by the mesh signs.
Does a larger driven gear increase or reduce speed?
A larger driven gear gives i greater than 1, so it reduces output speed and increases ideal torque magnitude. A smaller driven gear increases speed and reduces ideal torque.
Do external gears rotate in the same direction?
No. Two meshing external gears on parallel shafts rotate in opposite directions when viewed from the same end. An external pinion and internal ring rotate in the same direction.
Does an idler gear change the ratio?
A simple idler does not change the endpoint ratio magnitude because its tooth count cancels. It adds a mesh, so it can change the output direction. A compound intermediate pair does affect ratio.
How do I calculate a compound gear ratio?
Multiply the driven-to-driver ratio for every stage. For stages of 3:1 and 4:1, the overall reduction is 12:1. Multiply stage efficiency values separately.
How does gear ratio affect torque?
Ideal output torque magnitude equals input torque multiplied by the reduction factor. For moving forward power flow, an estimate with losses is Tout = Tin times i times efficiency.
What is the planetary gear speed formula?
For a simple sun-ring-carrier set, Zs times sun speed plus Zr times ring speed equals (Zs + Zr) times carrier speed. All speeds need one signed direction convention.
How is gear centre distance calculated?
For standard external spur gears, centre distance is half the sum of pitch diameters. For an internal pair, it is half the ring-minus-pinion pitch-diameter difference.
Does this calculator select a safe gear or gearbox?
No. It does not check tooth stress, contact fatigue, interference, shafts, bearings, lubrication, thermal limits, guards, shock loads or service factors. Verify a real design independently.
Method and Review Basis
Fixed-axis ratios and gear-train multiplication were checked against MIT gear-train notes and NASA Gearing. Spur reference geometry and internal-gear limits were checked against KHK basic gear calculations and KHK gear systems. Planetary equations were checked against MIT planetary gear-train notes. Rotational power and units were checked against OpenStax rotational power and the NIST SI conversion guide. Guarding scope was reviewed against OSHA mechanical power-transmission requirements.
Educational and machine-safety disclaimer: This calculator provides idealized kinematic results and user-assumption torque estimates. It does not certify gear geometry, strength, durability, lubrication, thermal performance, guarding or machine safety. Confirm inputs, units, operating direction, load cases, standards and manufacturer data. Have qualified professionals review consequential designs and isolate machinery before inspection.