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Gear Ratio Calculator

Advanced transmission physics tool. Calculate multi-stage gear trains, bicycle gear inches, skid patches, and robotic torque. Includes massive database of Shimano/SRAM components.

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Category Physics & Science
Drivetrain Configuration
Comma separated values
Comma separated values
Multi-Stage Train Builder
Total Ratio -- : 1
Output Speed -- RPM
Output Torque -- Nm
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Input
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Output
Animation speed is relative. Real RPM: 0
Gear Parameter Converter

Convert between Metric Module (m), Imperial Diametral Pitch (DP), and Circular Pitch (CP).

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About

Transmission design determines the operational envelope of any mechanical system. Whether optimizing a cyclist's cadence for a 15% gradient or calculating the holding torque of a robotic arm, the relationship between input speed and output force is governed by the laws of mechanical advantage. This tool goes beyond simple division, accounting for multi-stage compound trains, system efficiency losses, and specific cycling metrics like Gain Ratio and Skid Patches.

Precision is paramount. In robotics, a miscalculated ratio can lead to motor stall or thermal runaway. In cycling, understanding "Development Meters" allows for precise cassette selection, ensuring the rider maintains optimal physiological efficiency (zones). This calculator integrates standard ISO/ETRTO wheel sizes and industry-standard component libraries (Shimano, SRAM, Campagnolo) to provide actionable engineering data.

gear ratio torque calculator cycling physics skid patches robotics transmission mechanical engineering gear inches

Formulas

The core mechanical advantage is derived from the tooth counts z. For a simple pair, the ratio i is:

i = zdrivenzdrive

For a multi-stage (compound) train with n stages, the total ratio is the product of individual stages:

itotal = n∏k=1 zout,kzin,k

To calculate the output torque τout considering efficiency η (e.g., 0.95 for steel gears):

τout = τin × itotal × ηn

Cycling: Skid Patches (Fixed Gear)

SP = zchainringgcd(zchainring, zcog)

Note: If the rider is ambidextrous (skids with both legs), the value is × 2 if the numerator is odd.

Reference Data

ParameterSymbolUnitDefinition / Formula
Gear Ratioi:1The ratio of input angular velocity to output angular velocity. NdrivenNdrive
TorqueτNmRotational force. Increases with gear reduction. τout ≈ τin × i
Speed (Angular)ωRPMRotational speed. Decreases with gear reduction. ωout = ωini
Gear InchesGIinDiameter of an equivalent direct-drive wheel. Historical cycling metric.
DevelopmentDmDistance traveled per one crank revolution. D = Cwheel × i-1
Gain RatioGRratioRatio of wheel travel distance to pedal travel distance. Accounts for crank length.
Skid PatchesSPcountNumber of distinct contact points on a rear tire for fixed-gear braking.
ModulemmmRatio of Pitch Diameter to Number of Teeth. m = dz

Frequently Asked Questions

Gear Inches is an imperial unit representing the diameter of the wheel if the bicycle were a penny-farthing (direct drive). It does not directly tell you distance. Development Meters (or Roll-out) is a metric unit representing exactly how many meters the bike travels for one full rotation of the pedals. Development is generally preferred in modern sports science for its direct correlation to speed.
Theoretical calculations often ignore "Parasitic Loss". Every gear mesh interface has friction, and bearings add drag. A standard spur gear stage is about 95-98% efficient. A worm gear can be as low as 50%. This tool includes an "Efficiency" field to account for these thermodynamic losses.
When braking on a fixed gear (locking the legs), the rear wheel stops at specific points relative to the crank position. If your gear ratio creates only 1 skid patch, you will burn through your tire at that single spot very quickly. A higher number (e.g., 17 or 19 patches) distributes wear evenly around the tire circumference.
Proposed by Sheldon Brown, the Gain Ratio is the only unit that accounts for Crank Arm Length. Two bikes with the same gearing and wheels will feel different if one has 170mm cranks and the other 175mm. The Gain Ratio is a pure dimensionless number representing the distance the bike moves divided by the distance the pedal axle moves.
Yes. In the "Robotics & Engineering" tab, you can select the planetary mode. The ratio depends on which component is fixed (Ring, Sun, or Carrier). For a standard reduction (Ring fixed, Sun input, Carrier output), the ratio is 1 + (RingTeeth / SunTeeth).