Gear Ratio Calculator

Analyze any two‑gear system: compute gear ratio, output RPM, output torque, and direction. Visualize driver/driven gears with proportional sizing and rotation arrows.

Pinion / input gear
Output gear
✓ Teeth must be positive integers (realistic gears). Ideal efficiency (100%) assumed for perfect meshing – real systems have 95‑98% efficiency.
⛰️ Mountain low (32/48)
? Overdrive (50/30)
?️ Car 1st gear (12/45)
⚙️ Direct drive (20/20)
? High cadence (44/16)
? Reducer (15/60)
Precision local computation – All gear calculations are performed inside your browser. The interactive drivetrain drawing uses HTML5 canvas, no server-side storage.

Core principles: Gear ratio and mechanical advantage

The gear ratio (i) defines the relationship between two meshing gears: i = N₂ / N₁ where N₂ = number of teeth on driven gear, N₁ = number of teeth on driver gear. This ratio directly influences output speed and output torque:

ω₂ = ω₁ / i  and  T₂ = T₁ × i (ideal, lossless)

If i > 1, the system acts as a torque multiplier (speed reduction). If i < 1, it's an overdrive (speed increase, torque reduction). In real-world applications, friction reduces efficiency, but our calculator provides the theoretical baseline used by engineers for initial design.

Automotive & heavy machinery

Transmissions use multiple gear ratios to balance acceleration and fuel economy. Low gears (high ratio) multiply torque for climbing or towing; overdrive gears reduce RPM at highway speeds, enhancing efficiency.

Bicycle drivetrains

Chainrings (front) and cassette sprockets (rear) form a gear train. A larger rear sprocket provides easier climbing (high mechanical advantage), while a smaller rear sprocket increases speed per pedal revolution.

Why gear ratio matters – real‑world case study

Industrial conveyor system

A motor delivers 1450 RPM and 120 Nm torque. To drive a heavy roller at 350 RPM, required ratio = 1450/350 ≈ 4.14. Using a driver with 22 teeth and driven with 91 teeth (ratio 4.136), output torque becomes 120 × 4.136 ≈ 496 Nm, sufficient to move bulk materials. This calculator validates such configurations instantly, saving hours of manual calculation.

Compound gear trains & idler gears

In more complex systems, multiple gear pairs are combined: overall ratio = product of individual ratios. An idler gear between driver and driven does not affect the total ratio but reverses direction or spans distances. Our tool focuses on fundamental 2‑gear analysis, which is the building block for all epicyclic and planetary systems.

Step‑by‑step calculation process

  1. Enter driver teeth (N₁) and driven teeth (N₂) – positive integers required.
  2. Specify input RPM and input torque (any positive real numbers).
  3. Click Calculate & Visualize → gear ratio computed, output RPM = input RPM / i, output torque = input torque × i.
  4. Interactive canvas draws two proportional gears: radius scales with tooth count (within dynamic bounds). Direction arrows indicate opposite rotation (for external spur gears).

Quick reference: common gear configurations

Application Driver / Driven teeth Gear ratio (i) Output characteristic
Electric screwdriver 10 / 50 5.00 High torque, low speed
Bicycle top gear 48 / 11 0.229 High speed, low torque
Automotive 3rd gear 28 / 38 1.357 Balanced performance
Robotic joint actuator 12 / 60 5.00 Precise torque control
Wind turbine gearbox 19 / 95 5.00 Multi‑stage speed step‑up

Euler‑Eytelwein & beyond: historical perspective

Gear theory matured during the Industrial Revolution. In the 18th century, Euler derived involute tooth profiles, while modern mechanical engineers apply the fundamental law of gearing: the angular velocity ratio must remain constant. The orthocenter of a triangle and gear ratios share a mathematical elegance – both rely on proportional relationships. Today, gear design software and interactive calculators democratize this knowledge, enabling rapid prototyping.

Frequently Asked Questions

A 2:1 ratio (driver:driven teeth 20:40) means the driven gear rotates once for every two driver rotations. Output torque doubles, output speed halves.

Gear teeth must be integer to maintain proper meshing and uniform angular transmission. Fractional teeth are physically impossible in standard involute gears.

No, idler gears only reverse rotation direction. The overall ratio between input and output remains N₂/N₁, independent of the idler tooth count.

Ideal torque multiplication ignores friction. In real systems, efficiency (95‑98% for spur gears) slightly reduces output torque. For preliminary design, the ideal value is standard.

No, the calculator now enforces integer teeth. Please use whole numbers (e.g., 20, 45) for physically realistic gearing.

Internal gears produce same‑direction rotation, but our visualization focuses on external spur gears (most common). The ratio formula remains valid.
Engineering references & expertise – Formulas validated using AGMA (American Gear Manufacturers Association) standards. Data derived from "Shigley's Mechanical Engineering Design" and "Dudley's Gear Handbook".Last verification: June 2026.

Created by mechanical design team – merging interactive simulation with classical gear theory. All calculations run locally; no input leaves your device. Transparent, reliable, and classroom‑ready.