Gear Ratio Calculator
Calculate gear ratio calculator calculate gear ratio GR = Ndriven / Ndriver, output speed omega_out = omega_in / GR, and output torque tau_out = tau_in * GR from teeth counts, RPM, and torque. Accurate physics formulas and unit conversions for engineers, students & technicians.
Calculate Gear Train Kinematics (GR = N_driven / N_driver)
Enter your physical parameters below to compute verified gear ratio metrics.
Calculation Results
Calculated using verified physical methodology: Gear Velocity Ratio: GR = \frac{N_{\text{driven}}}{N_{\text{driver}}} = \frac{D_{\text{driven}}}{D_{\text{driver}}}
Output Speed: \omega_{\text{out}} = \frac{\omega_{\text{in}}}{GR}
\text{Output Torque: } \tau_{\text{out}} = \tau_{\text{in}} \cdot GR
Quick Summary
The Gear Ratio Calculator evaluates gear ratio ($GR = \frac{N_{\text{driven}}}{N_{\text{driver}}}$), output speed ($\omega_{\text{out}} = \frac{\omega_{\text{in}}}{GR}$), and output torque ($\tau_{\text{out}} = \tau_{\text{in}} \cdot GR$) for spur gears, gearboxes, and chain sprockets.
Formula Explanation
Gear Velocity Ratio: GR = \frac{N_{\text{driven}}}{N_{\text{driver}}} = \frac{D_{\text{driven}}}{D_{\text{driver}}}
Output Speed: \omega_{\text{out}} = \frac{\omega_{\text{in}}}{GR}
\text{Output Torque: } \tau_{\text{out}} = \tau_{\text{in}} \cdot GR
How It Works
The Gear Ratio Calculator divides driven gear teeth ($N_{\text{driven}}$) by driver gear teeth ($N_{\text{driver}}$) to compute $GR$. It divides input speed ($\omega_{\text{in}}$ in RPM) by $GR$. It multiplies input torque ($\tau_{\text{in}}$) by $GR$. It outputs gear ratio ($GR:1$), output RPM, output torque in N·m and ft-lbs, and mechanical power ($\text{kW} = \2\pi \tau \omega / 60,000$).
Step-by-Step Worked Example
Practical Problem: An electric motor running at $\omega_{\text{in}} = 1,800.0\text{ RPM}$ with torque $\tau_{\text{in}} = 50.0\text{ N}\cdot\text{m}$ drives a $N_{\text{driver}} = 12\text{ teeth}$ pinion gear meshed to a $N_{\text{driven}} = 48\text{ teeth}$ reduction gear. Calculate gear ratio $GR$, output speed $\omega_{\text{out}}$, and output torque $\tau_{\text{out}}$.
- Step 1: Identify Input Parameters: $N_{\text{driver}} = 12$, $N_{\text{driven}} = 48$, $\omega_{\text{in}} = 1,800.0\text{ RPM}$, $\tau_{\text{in}} = 50.0\text{ N}\cdot\text{m}$.
- Step 2: Calculate Gear Ratio ($GR = N_{\text{driven}} / N_{\text{driver}}$): $GR = \48 / 12 = 4.00\text{ (4.00:1 gear reduction)}$.
- Step 3: Calculate Output Driven Speed ($\omega_{\text{out}} = \omega_{\text{in}} / GR$): $\omega_{\text{out}} = \frac{1,800.0\text{ RPM}}{4.00} = 450.00\text{ RPM}$.
- Step 4: Calculate Output Driven Torque ($\tau_{\text{out}} = \tau_{\text{in}} \cdot GR$): $\tau_{\text{out}} = 50.0\text{ N}\cdot\text{m} \times 4.00 = 200.00\text{ N}\cdot\text{m}$.
- Step 5: Calculate Mechanical Power Conservation: Power $P = \2\pi \times 50 \times 1800 / 60,000 = 9.425\text{ kW}$ ($12.63\text{ hp}$). Output power equals input power ($450\text{ RPM} \times 200\text{ N}\cdot\text{m} = 9.425\text{ kW}$).
Real-World Calculation Examples
Scenario 1: Industrial Speed Reducer Gearbox (12T -> 48T @ 1800 RPM)
Parameters: $N_{\text{driver}} = 12$, $N_{\text{driven}} = 48$, $\omega_{\text{in}} = 1,800\text{ RPM}$, $\tau_{\text{in}} = 50\text{ N}\cdot\text{m}$
Result: $GR = 4.00:1$, $\omega_{\text{out}} = 450.00\text{ RPM}$, $\tau_{\text{out}} = 200.00\text{ N}\cdot\text{m}$. 4x torque gain.
Scenario 2: Automobile Differential Final Drive (11T pinion -> 41T ring)
Parameters: $N_{\text{driver}} = 11$, $N_{\text{driven}} = 41$, $\omega_{\text{in}} = 3,000\text{ RPM}$, $\tau_{\text{in}} = 250\text{ N}\cdot\text{m}$
Result: $GR = 3.727:1$, $\omega_{\text{out}} = 804.88\text{ RPM}$, $\tau_{\text{out}} = 931.82\text{ N}\cdot\text{m}$. Car rear axle differential.
Scenario 3: Bicycle Overdrive (44T chainring -> 11T rear cog)
Parameters: $N_{\text{driver}} = 44$, $N_{\text{driven}} = 11$, $\omega_{\text{in}} = 90\text{ RPM}$ (pedaling cadence)
Result: $GR = 0.25:1$ ($1:4.00$ overdrive), $\omega_{\text{out}} = 360.00\text{ RPM}$. High-speed sprinting gear.
Scenario 4: Robotics Compound Servo Gearhead (10T -> 100T)
Parameters: $N_{\text{driver}} = 10$, $N_{\text{driven}} = 100$, $\omega_{\text{in}} = 6,000\text{ RPM}$, $\tau_{\text{in}} = 0.5\text{ N}\cdot\text{m}$
Result: $GR = 10.00:1$, $\omega_{\text{out}} = 600.00\text{ RPM}$, $\tau_{\text{out}} = 5.00\text{ N}\cdot\text{m}$. Robotic actuator torque multiplier.
Key Benefits of Using This Calculator
Speed vs Torque Tradeoff
Demonstrates how gear reduction ($GR > 1$) increases output torque ($\tau_{\text{out}} = \tau_{\text{in}} \cdot GR$) while proportionally reducing output speed ($\omega_{\text{out}} = \omega_{\text{in}}/GR$).
Power Conservation Integration
Computes mechanical power ($P = \2\pi\tau\omega / 60,000$ in kW & hp) to verify energy conservation across gear meshing.
Multi-Unit Readouts
Outputs gear ratio ($GR:1$), output RPM, output torque in N·m and ft-lbs, and shaft power.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a gear ratio?
Gear ratio (GR) is the direct ratio of the number of teeth on the driven (output) gear to the number of teeth on the driver (input) gear (GR = Ndriven / Ndriver).
What is gear reduction vs gear overdrive?
Gear Reduction (GR > 1): Small driver drives large driven (increases torque, decreases speed); Overdrive (GR < 1): Large driver drives small driven (increases speed, decreases torque).
How relates gear pitch diameter to teeth count?
For meshing gears, module or diametral pitch must be identical, so Diameter D is directly proportional to teeth count N: GR = Ndriven / Ndriver = Ddriven / Ddriver.
What is the role of an idler gear?
An idler gear inserted between driver and driven gears changes the DIRECTION of output rotation without altering the overall gear ratio (GR = Ndriven / Ndriver regardless of idler teeth!).
How do you calculate compound gear train overall ratio?
Overall Gear Ratio GR_total = GR1 * GR2 * GR3 = (Ndriven1 / Ndriver1) * (Ndriven2 / Ndriver2)...
How converts output N*m torque to Imperial foot-pounds (ft-lbs)?
Multiply N*m by 0.737562 (1 N*m = 0.737562 ft-lbs).
What is hunting tooth gear ratio?
A hunting tooth design uses coprime teeth numbers (e.g. 13 and 47) so that individual teeth mesh evenly across all opposite teeth, preventing localized wear patterns.
What is planetary / epicyclic gear train?
A planetary gear set consists of a sun gear, planet gears, and outer ring gear, providing compact ultra-high gear reductions inside automatic car transmissions.
How does chain sprocket ratio work on bicycles and motorcycles?
Identical to spur gears! Chain Sprocket Ratio = N_driven_sprocket / N_driver_sprocket.
Why does output torque increase when output RPM decreases?
By Conservation of Power: Power P = Torque * Angular Velocity = constant; if speed decreases by 4x, torque MUST increase by 4x to maintain energy conservation.