AtlasCalcsUtility Calculation Suite
August 2026 12 min read

Gear Ratio Calculator

Mechanical gear ratio calculator: compute the ratio from tooth counts, output speed and torque, pitch diameters, center distance and pitch-line velocity for spur and helical gear pairs.

Gear Ratio Calculator

Gear Pair Inputs

Unit System
SI — mm, N·m Imperial — in, lbf·in
Driver Teeth T₁ (T)

Teeth on the input gear

Driven Teeth T₂ (T)

Teeth on the output gear (sets the ratio)

Module m (mm)

Tooth size (same for both gears)

Input Speed (rpm)

Driver rotational speed

Input Torque (N·m)

Driver torque

Solve For

Which output to emphasize (both are shown)

Auto-calculate instantly

Output Speed

750rpm

Gear ratio 2:1, input 1,500 rpm.

Pitch-line velocity is modest — quiet running
Gear Ratio

2:1

Driver Pitch Ø

40mm

Driven Pitch Ø

80mm

Center Distance

60mm

Output Speed

750rpm

Output Torque

100N·m

Pitch Vel (driver)

3.14m/s

Pitch Vel (driven)

3.14m/s

Gear Ratio

2:1

T₂ / T₁

Driver Pitch Ø

40mm

m·T₁

Driven Pitch Ø

80mm

m·T₂

Center Distance

60mm

(d₁+d₂)/2

Output Speed

750rpm

n_in / ratio

Output Torque

100N·m

T_in × ratio

Pitch Vel (driver)

3.14m/s

π·d₁·n/60000

Pitch Vel (driven)

3.14m/s

π·d₂·n_out/60000

Result Summary

Driver Teeth20 T
Driven Teeth40 T
Module2 mm
Gear Ratio2 : 1
Center Distance60 mm
Output Speed750 rpm
Output Torque100 N·m
Pitch Line Velocity3.14 m/s
Estimated Result — Assumes standard external spur/helical gears with equal module and ideal (lossless) meshing. Ratio, pitch diameters, center distance and pitch-line velocity are exact kinematics; real torque includes mesh efficiency (≈97–99% per stage) and the tooth strength must be checked separately (AGMA/ISO 6336). For worm, bevel, planetary or compound trains, combine multiple stages or use the dedicated calculator.

Interactive Diagram

A driver (20T) and a driven (40T) gear give a 2:1 reduction: half the speed, twice the torque.

Visualizations

Gear Pair Geometry

40d₁
80d₂
60a
1,000T_out
750n_out

Pitch-Line Velocity (driver)

v = π·d₁·n/60000310.1 m/s

Formula

A gear pair changes speed and torque by the tooth-count ratio. The pitch diameters and center distance follow from the module (tooth size). Pitch-line velocity sets the noise, wear and precision class. For a compound train the overall ratio is the product of the stage ratios.

GR = T₂ / T₁

Alternative Form

n_out = n_in / GR , T_out = T_in × GR , a = m(T₁+T₂)/2 , v = π·d·n/60000

Variables

SymbolNameUnitDescription
T₁, T₂Teeth counts—Driver and driven teeth
mModulemm/toothMust match between meshing gears
dPitch diametermmd = m·T
aCenter distancemm(d₁+d₂)/2
nSpeedrpmRotational speed
TTorqueN·mTwisting effort
vPitch-line velocitym/sπ·d·n/60000

Explanation

Gears trade speed for torque. A reduction (GR > 1) makes the output slower but stronger; an overdrive (GR < 1) does the opposite. The kinematic ratio is exact, but the real output torque is reduced by mesh efficiency (about 1–3% loss per stage). The pitch-line velocity v = π·d·n/60000 tells you whether the pair can run quietly: below ~10 m/s cast/spur gears are fine, above ~20 m/s you need ground helical gears and good lubrication.

Derivation

From rolling without slip at the pitch point, n₁T₁ = n₂T₂, so n₂/n₁ = T₁/T₂ and GR = T₂/T₁. Pitch diameter d = mT and for an external pair the center distance is half the sum of pitch diameters. Pitch-line velocity v = ω·r = (2πn/60)·(d/2) = π·d·n/60000 (with d in mm and n in rpm).

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Compute the gear ratio from tooth counts.

GR = T₂ / T₁
T₁:20
T₂:40
Intermediate: GR = 40 / 20
Result: 2 : 1
Step 2

Find the pitch diameters for the chosen module.

d = m·T
m:2 mm
Intermediate: d₁ = 2 × 20 = 40 mm
Result: d₂ = 80 mm
Step 3

Set the center distance.

a = (d₁ + d₂) / 2
d₁:40 mm
d₂:80 mm
Intermediate: a = (40 + 80) / 2
Result: 60 mm
Step 4

Apply the ratio to speed and torque.

n_out = n_in / GR , T_out = T_in × GR
n_in:1,500 rpm
T_in:50 N·m
Intermediate: n_out = 1,500 / 2
Result: 750 rpm, 100 N·m
Step 5

Compute pitch-line velocity for noise/wear.

v = π·d·n / 60000
d₁:40 mm
Intermediate: v = π × 40 × 1,500 / 60000
Result: 3.14 m/s
Final Answer
750 rpm

Worked Examples

See how the calculator works with different scenarios.

Scenario

A 20-tooth driver, 40-tooth driven, module 2 mm.

Inputs

T₁:20 T
T₂:40 T
m:2 mm
n_in:1500 rpm
T_in:50 N·m

Calculation

GR = 40/20 = 2; d₁ = 40 mm, d₂ = 80 mm, a = 60 mm; n_out = 1500/2 = 750 rpm; T_out = 50×2 = 100 N·m; v = π×40×1500/60000 = 3.14 m/s

Result

2:1 reduction → 750 rpm, 100 N·m

Interpretation

Classic halving of speed with doubling of torque (ignoring losses).

Understanding Your Result

Your Result: 750 rpm

Typical range: Single-stage spur/helical ratios are practical up to about 6–8:1. Your 2:1 is within a single-stage range.

What This Means

This 20T → 40T pair gives a 2:1 ratio. Output speed is 750 rpm, output torque is 100 N·m, center distance 60 mm, pitch-line velocity 3.14 m/s.

Why It Matters

Gear ratio is the first decision in any transmission: it sets whether a motor can deliver the needed speed and torque. Get it wrong and the machine is either too slow to be useful or so torque-starved it stalls. Center distance and pitch velocity then drive the physical layout and the noise/wear regime.

Professional Interpretation

Verify tooth bending and pitting capacity (AGMA/ISO 6336) for the actual load, especially with pitch velocity 3.14 m/s. Account for mesh efficiency (≈97–99%/stage) on torque, and keep T₁ above the undercut minimum (~17 at 20° pressure angle) or use profile shift.

Velocity class

v = 3.14 m/s → low (quiet).

Torque multiplication

T_out = 100 N·m = 2 × input.

Compound trains

Overall ratio = product of stage ratios; allows very high reduction compactly.

Recommendations

  • •Single-stage ratio is reasonable.
  • •Keep center distance integer-friendly by choosing module and tooth counts so (T₁+T₂)·m/2 is manufacturable.
  • •Below 10 m/s, standard spur gears are fine; above 20 m/s use ground helical gears and forced lubrication.
  • •Multiply output torque by ~0.97–0.99 per mesh for real (lossy) drive torque.
  • •Check tooth bending (AGMA) separately — kinematics alone do not guarantee strength.

Educational Content

Introduction

A gear ratio is the simplest, most powerful tool in mechanical design: it tells you exactly how a gear pair will change speed and torque. This calculator takes the tooth counts and module of a spur or helical pair and returns the ratio, the resulting output speed and torque, the pitch diameters, the center distance and the pitch-line velocity. With those you can lay out a drive, pick a motor, or sanity-check a transmission.

Concept Explanation

The gear ratio GR = T₂/T₁ is the tooth-count ratio between driven and driver. Because the pitch circles roll without slipping, the speed ratio is the inverse of the tooth ratio and the torque ratio is directly GR (times efficiency). The pitch diameter d = m·T sets the physical size, the center distance a = m(T₁+T₂)/2 sets the shaft spacing, and the pitch-line velocity v = π·d·n/60000 sets the noise and wear regime. For compound trains, the overall ratio is the product of each stage's ratio.

Applications

▸Sizing a gear reduction for a motor
▸Designing a transmission or gearbox
▸Picking center distance and shaft layout
▸Estimating noise/wear class from pitch velocity
▸Teaching velocity and torque trade-offs
▸Checking idler and compound arrangements

Benefits

  • •Exact kinematic ratio, speed and torque from simple inputs
  • •Returns pitch diameters, center distance and pitch velocity
  • •Flags high-velocity regimes that need precision gears
  • •Handles reduction and overdrive
  • •SI / imperial consistent

Limitations

  • •Assumes ideal, lossless meshing (no efficiency factor applied to torque)
  • •One external pair only (worm/bevel/planetary/compound need extension)
  • •Does not check tooth bending or pitting strength
  • •Equal module assumed (must hold for meshing gears)
  • •No profile shift or undercut handling

Best Practices

  • •Keep the driven teeth count and module such that center distance is a clean, machinable value
  • •Use helical or ground gears above ~20 m/s pitch velocity
  • •Apply ~0.97–0.99 per mesh for real output torque
  • •Keep T₁ above the undercut minimum or use profile shift
  • •Validate with AGMA/ISO 6336 for the actual transmitted load

Common Misconceptions

Myth: A bigger gear always slows things down.
Truth: Only if it is the driven gear; an idler (equal ratio) changes spacing/direction, not speed.
Myth: Gear ratio multiplies torque for free.
Truth: Power is conserved (minus losses); more torque means proportionally less speed.
Myth: Ratio is about diameters, not teeth.
Truth: For equal-module gears diameters scale with teeth, so T₂/T₁ = d₂/d₁ — teeth are the practical input.
Myth: Pitch velocity does not matter.
Truth: Above ~10 m/s noise and wear rise sharply; design and lubrication must change.

Practical Tips

  • •Ratio 2:1 → output 750 rpm, 100 N·m.
  • •For very high ratios, split into a compound train rather than one huge gear.
  • •Keep pitch-line velocity below ~10 m/s for quiet cast spur gears.
  • •Multiply output torque by ~0.98 per mesh for real drive torque.
  • •Choose module for tooth strength first, then set teeth for the ratio.

Common Mistakes

Mismatched module
Both gears must share the same module; otherwise teeth will not mesh.
Forgetting efficiency
Real torque = input × ratio × 0.97–0.99 per mesh, not ratio alone.
One-stage ratio too high
A 10:1 single stage makes a huge driven gear; use compound/planetary.
Ignoring pitch velocity
Above ~20 m/s use ground helical gears and forced lubrication.
Undercut teeth
Keep T₁ ≥ ~17 at 20° pressure angle, or apply profile shift.

Assumptions

This calculator makes the following assumptions to provide estimates:

Ideal meshing: Rolling without slip; efficiency loss not applied to torque.
Equal module: Both gears share module m so teeth mesh.
External spur/helical pair: Parallel shafts, external mesh.
Standard pressure angle: 20° assumed for undercut guidance; profile shift not modelled.
No strength check: Kinematics only; AGMA/ISO durability is separate.

Limitations

Be aware of these limitations when using this calculator:

Single pair: Compound, worm, bevel and planetary trains need extension.
No efficiency: Torque is lossless; apply η per mesh for reality.
No load rating: Tooth bending/pitting requires AGMA/ISO 6336.
No profile shift: Undercut avoidance by minimum teeth only.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a gear ratio?
The ratio of the driver gear's teeth to the driven gear's teeth. A 20-tooth driver and a 40-tooth driven give a ratio of 2:1, meaning the output turns half as fast and with twice the torque.
How do I calculate gear ratio?
Gear ratio = T_driven / T_driver (or D_driven / D_driver for same-module gears). For speed: n_out = n_in / ratio; for torque: T_out = T_in × ratio (ignoring losses).
What is a compound gear train?
Two or more gear pairs on shared shafts. The overall ratio is the product of each stage's ratio, allowing very high reductions in a compact space.
What is the difference between reduction and overdrive?
Reduction (ratio > 1) slows the output and raises torque; overdrive (ratio < 1) speeds the output and lowers torque. Most machinery uses reduction.
What is module in gears?
Module m = pitch diameter / number of teeth (mm/tooth). Both meshing gears must share the same module so the teeth fit. It sets tooth size.
How do I find center distance?
For a pair: a = m(T1 + T2) / 2. For a parallel-axis external gear pair this is simply half the sum of pitch diameters.
What is pitch line velocity?
The speed at the pitch circle: v = π·d·n / 60 (m/s). It sets noise, wear and the required precision/balance class.
What does a gear ratio of 1:1 mean?
Both gears have the same number of teeth, so speed and torque are unchanged (only direction reverses for external gears).
How does gear ratio affect torque?
Torque scales directly with the ratio: a 3:1 reduction triples the output torque. This is why low gears in a car produce high wheel torque.
How does gear ratio affect speed?
Output speed is inversely proportional to the ratio: a 4:1 reduction makes the output turn at one quarter the input speed.
What is the relationship with RPM?
n_out = n_in × (T_in / T_out). More teeth on the driven gear means slower output.
What is backlash in gears?
The small gap between meshing tooth flanks that allows free motion and compensates for thermal growth. Too much causes lost motion and noise.
What are the types of gears?
Spur (straight teeth, parallel shafts), helical (angled, smoother), bevel (intersecting shafts), worm (high reduction, right-angle) and planetary (compact, high ratio).
What is a worm gear ratio?
Often set by the worm's starts and the wheel's teeth: ratio = T_wheel / starts. A single-start worm gives very high reduction and self-locking.
What is a planetary gear set?
Sun, planets, ring and carrier. Selecting which member is fixed/input/output gives many ratios in a compact, coaxial package — common in automatics and robotics.
How do I calculate efficiency?
Each mesh loses 1–3% (spur/helical). Overall η = product of stage efficiencies. Worm gears can be as low as 50–90% depending on lead angle.
What is the velocity ratio?
The exact kinematic ratio n_in/n_out = T_out/T_in, independent of tooth profile; what the gear ratio gives.
How do I choose a gear ratio?
Match the load's required speed and torque to the motor's, considering efficiency, space, ratio range and whether you need reduction or overdrive.
What is the maximum practical gear ratio for one stage?
Spur/helical single stages are practical up to about 6–8:1; beyond that use compound, worm or planetary stages to keep sizes reasonable.
What is the law of gearing?
For constant velocity ratio, the common normal at the contact point must always pass through the fixed pitch point (the fundamental law of conjugate action).
What is pressure angle?
The angle between the tooth force and the tangent at the pitch point (typically 20°). It affects strength, smoothness and minimum tooth count before undercut.
What is undercut in gears?
Weak material removed at the tooth root when too few teeth are used at a given pressure angle, weakening the tooth. Avoided with enough teeth or profile shift.
What is the contact ratio?
The average number of tooth pairs in contact. A value ≥ 1.2 ensures smooth, continuous meshing without gaps.
How do helical gears differ in ratio?
The ratio is still T_driven/T_driver, but the helix angle introduces axial thrust and smoother, quieter meshing.
What is gear module vs diametral pitch?
Module (mm/tooth, metric) and diametral pitch (teeth/inch, imperial) are reciprocally related via π; both size the teeth so mating gears align.
How do I reverse rotation direction?
An external gear pair reverses direction; an idler gear keeps the same direction but changes spacing. Internal gears also preserve direction.
What is an idler gear?
A gear between driver and driven that only changes direction or spacing; it does not change the overall ratio (it cancels out).
What is torque multiplication?
The useful effect of reduction: output torque = input torque × ratio × efficiency. It is the reason gears are used in transmissions.
How do I size a gear for a given ratio?
Pick a module for the load (tooth strength), then choose teeth counts T1 and T2 with T2/T1 = ratio and T1 above the undercut minimum.
What is the difference between ratio and mechanical advantage?
Ratio is kinematic (speed/torque change by the numbers); mechanical advantage includes efficiency losses, so real torque gain is ratio × η.
What is a differential gear?
A gear set that splits torque and allows two outputs (wheels) to rotate at different speeds while sharing torque — not a fixed ratio component.
How do gear ratios affect fuel economy?
Taller (lower numerical) final drive ratios keep engine rpm lower at cruise, improving economy; shorter ratios improve acceleration.
What is the AGMA rating for gears?
AGMA provides rating methods for gear tooth strength (bending) and surface durability (pitting) used alongside the kinematic ratio.
Can gear ratio be a decimal?
Yes — e.g., 0.75:1 (overdrive) or 3.55:1 (reduction). The calculator accepts any positive tooth or speed values.
How do I calculate gear ratio from RPM?
ratio = n_in / n_out. If input is 3000 rpm and output 1000 rpm, the ratio is 3:1.
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Glossary

Technical terms used in mortgage calculations explained in plain language.

Gear Ratio (GR)
T_driven / T_driver; output speed factor.
Module (m)
Pitch diameter per tooth (mm/tooth); must match between meshing gears.
Pitch Diameter (d)
d = m·T; the reference rolling diameter.
Center Distance (a)
a = m(T1 + T2)/2 for an external pair.
Pitch Line Velocity (v)
v = π·d·n/60; surface speed at the pitch circle.
Velocity Ratio
n_in / n_out = T_out / T_in (kinematic).
Compound Train
Multiple stages; overall ratio = product of stage ratios.
Efficiency (η)
Power out / power in; each mesh loses 1–3%.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

AGMAISO 6336ISO 1328DIN 3975Shigley / machine-design texts

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Page Information

Last Updated:

August 2026

Version:

2.1.0

License:

Free to use for educational and professional purposes.