What is the formula for gear ratio?
The gear ratio (GR) for a simple pair of meshing gears is GR = teeth on the driven gear ÷ teeth on the driving gear. Equivalently, GR = N_out ÷ N_in, where N denotes tooth count. In ideal (lossless) conditions, torque_out = torque_in × GR and speed_out = speed_in ÷ GR. Below, we clarify conventions, show equivalent forms, and provide examples and tips to avoid common mistakes.
Contents
What “gear ratio” means and why it matters
Gear ratio quantifies how a gear pair (or train) transforms speed and torque between an input (driver) and an output (driven) shaft. A ratio greater than 1 indicates reduction (slower output, higher torque), while a ratio less than 1 indicates overdrive (faster output, lower torque). Correctly interpreting the ratio helps in designing transmissions, bicycles, robotics gearboxes, industrial reducers, and EV drivetrains.
The core formula
For a single external gear mesh, with gear 1 as the driving gear and gear 2 as the driven gear:
GR = teeth_driven / teeth_driver = N₂ / N₁
In ideal conditions (no losses):
speed_out = speed_in / GR
torque_out = torque_in × GR
Because tooth pitch is common in a mesh, the same ratio applies to pitch diameters and radii:
GR = D₂ / D₁ = r₂ / r₁
Equivalent forms and sign conventions
The following points summarize equivalent ways engineers express gear ratio and how direction is handled in different references.
- Tooth-count form: GR = N_driven / N_driver
- Speed form (ideal): GR = ω_in / ω_out
- Torque form (ideal): GR = τ_out / τ_in
- Diameter form: GR = D_driven / D_driver
- External mesh reverses rotation direction; internal mesh preserves it. Some texts include a negative sign to indicate reversal (e.g., ω_out = −ω_in / GR), while others report only magnitude and note direction separately.
- “Reduction ratio” in product specs typically means GR ≥ 1, defined as input speed ÷ output speed.
In practice, always check the datasheet or textbook convention to confirm whether “gear ratio” is reported as driven/driver or the inverse, and whether sign is used to indicate direction.
Worked examples
Single pair
Driver: 20 teeth; Driven: 60 teeth. GR = 60/20 = 3. If the input shaft spins at 1500 rpm, output speed = 1500/3 = 500 rpm (opposite direction for external mesh). If input torque is 10 N·m, ideal output torque = 10 × 3 = 30 N·m.
Using diameters
Driver pitch diameter: 40 mm; Driven pitch diameter: 100 mm. GR = 100/40 = 2.5. Output speed = input speed / 2.5; output torque = input torque × 2.5 (ideal).
Compound and multi-stage gear trains
In multi-stage gear trains, the overall gear ratio is the product of each stage’s ratio. This allows large reductions or increases using several smaller meshes.
- Stage i ratio: GR_i = (teeth_driven,i) / (teeth_driver,i)
- Overall ratio: GR_total = GR_1 × GR_2 × … × GR_k
- Example: Stages with ratios 3, 2, and 1.5 give GR_total = 3 × 2 × 1.5 = 9
- Direction: Each external mesh flips direction. An even number of external meshes yields the same direction as input; an odd number flips it.
This product rule is essential when chaining spur gears, planetary stages, or mixed gear types to achieve a target speed and torque transformation.
Common pitfalls
These are frequent sources of confusion and how to address them.
- Mixing up driver and driven: Always label gears before calculating.
- Speed ratio vs gear ratio: Some references define “gear ratio” as driver/driven; verify context.
- Ignoring losses: Real systems have efficiency < 100%. Actual torque gain ≈ τ_in × GR × efficiency.
- Counting module or diametral pitch: Ratio depends on teeth count or pitch diameters, not tooth size alone.
- Directionality: External meshes reverse rotation; internal meshes do not.
Clarifying these points avoids design errors and mismatched expectations when selecting catalog gearboxes.
How to measure for a quick estimate
If tooth counts are unknown, you can still estimate the ratio with basic observations.
- Count teeth directly if accessible; it’s the most reliable quick method.
- Measure pitch diameters (or center distance and one diameter) to infer the ratio.
- Mark both shafts and time a fixed number of input turns to observe output turns; ratio ≈ input turns ÷ output turns.
These field methods provide workable estimates when documentation isn’t available.
Bottom line
The formula for gear ratio is GR = teeth on driven gear ÷ teeth on driving gear, with equivalent expressions in terms of speeds, torques, and diameters under ideal conditions. In multi-stage trains, multiply stage ratios. Always verify the convention used (which gear is “driven,” whether sign indicates direction, and whether “reduction ratio” is implied) and account for efficiency in real-world applications.
Summary
Gear ratio for a simple mesh: GR = N_driven / N_driver = ω_in / ω_out = τ_out / τ_in (ideal). Output speed equals input speed divided by GR; output torque equals input torque multiplied by GR. For multi-stage systems, multiply stage ratios. Confirm conventions and direction, and factor in efficiency for practical calculations.
How do you calculate the gear ratio?
To calculate a gear ratio, divide the number of teeth on the output (driven) gear by the number of teeth on the input (drive) gear. The formula is: Gear Ratio = Teeth on Output Gear / Teeth on Input Gear. For example, a gear with 32 teeth driving a gear with 8 teeth has a gear ratio of 4:1 (32/8).
This video demonstrates how to calculate gear ratio using the number of teeth: 43sEngineers AcademyYouTube · Feb 7, 2019
Steps to Calculate a Gear Ratio
- Identify the input and output gears: The input gear is the one that is being turned, and the output gear is the one that is being turned by the input gear.
- Count the teeth on each gear: Count the number of teeth on both the input gear and the output gear.
- Apply the formula: Divide the number of teeth on the output gear by the number of teeth on the input gear.
Understanding the Gear Ratio
- A ratio greater than 1 (e.g., 4:1): indicates a gear reduction, meaning the output gear turns slower than the input gear but provides greater torque.
- A ratio less than 1 (e.g., 1:2): indicates a gear overdrive, where the output gear turns faster than the input gear but with less torque.
- A ratio of 1:1: means both gears rotate at the same speed and have the same torque.
Other Methods (Equivalents)
The gear ratio can also be determined using other proportional measurements:
- Diameter or Radius: Gear Ratio = Diameter of Output Gear / Diameter of Input Gear OR Gear Ratio = Radius of Output Gear / Radius of Input Gear.
- Rotational Speed (RPM): Gear Ratio = Input RPM / Output RPM.
- Torque: Gear Ratio = Output Torque / Input Torque.
For Multiple Gear Pairs
If you have several gears in a series (a compound gear train), you can multiply the individual gear ratios together to find the overall gear ratio.
What is the 3.73 gear ratio?
“3.73 gears” refers to a gear ratio of 3.73:1, meaning the driveshaft must make 3.73 revolutions to turn the vehicle’s axle (and wheels) one full rotation. A higher gear ratio number like 3.73 provides more mechanical advantage, resulting in better acceleration and increased ability to tow or carry heavy loads, but at the cost of higher engine RPMs, lower top-end speed, and reduced fuel economy compared to a lower-numbered ratio.
What a 3.73 Gear Ratio Means
- Mechanical Advantage: It’s a higher ratio (or “shorter” gears), which gives the engine more torque to the wheels, similar to how a bicycle’s lowest gear helps you pedal up a steep hill.
- Engine Revolutions: The driveshaft spins 3.73 times for every one rotation of the rear axle.
- Torque and Acceleration: More torque means better off-the-line acceleration and a stronger ability to pull trailers or carry heavy loads.
The Pros and Cons of 3.73 Gears
- Pros:
- Improved acceleration.
- Better performance when towing heavy loads.
- More engine power to overcome hills or resistance.
- Cons:
- Lower fuel economy compared to lower gear ratios.
- Reduced top-end speed.
- Engine runs at higher RPMs for a given road speed.
When 3.73 Gears Are Ideal
- Trucks and SUVs: Opens in new tabFor vehicles that frequently tow heavy trailers or carry large payloads, the 3.73 ratio offers significant advantages.
- Performance Vehicles: Opens in new tabIn sports cars, a higher gear ratio like 3.73 can provide quicker acceleration from a stop.
When 3.73 Gears May Not Be Ideal
- Daily Driving: If your priority is fuel economy and comfortable highway cruising, a lower gear ratio (like 3.55) is generally better.
- Light Duty: For light towing or mostly city driving with an empty vehicle, the 3.73 gear set may be unnecessary and can hurt fuel efficiency.
What is the gear ratio in simple terms?
gear ratio = rotations of a driver gear : rotations of a driven gear. For every rotation of the 45-tooth gear, the 15-tooth gear must rotate 3 times. This is true no matter how many times the 45-tooth gear rotates. The ratio between the rotations of the 15-tooth driver gear and the 45-tooth driven gear is 3 to 1.
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What is a 20 to 1 gear ratio?
For instance, a NeveRest 20 gearmotor consists of an unmodified NeveRest Motor and a planetary gearbox that has a gear ratio of 20:1 (or, when spoken, “20 to 1”). This means that in order for the output shaft of the gearbox to rotate 1 time, the input shaft of the motor must rotate 20 times.
