Gear Ratio Calculator

Gear ratios determine how speed and torque are transferred between rotating shafts.

• A higher ratio means more torque but less speed, which is ideal for starting heavy loads.
• A lower ratio means more speed but less torque, which is ideal for high-speed operation.

Gear ratios are used in:

• automotive transmissions
• bicycles
• industrial machinery
• robotics
• power tools

Understanding ratios helps in selecting the right gears for your speed and torque requirements.

Gear ratio (GR) equals the number of teeth on the driven (output) gear divided by the teeth on the driving (input) gear: GR = N₂ / N₁. This ratio is dimensionless.

For example, a 15-tooth driving gear meshing with a 45-tooth driven gear gives GR = 45 / 15 = 3, written as 3:1.

Because gear teeth must interlock at the same pitch, teeth count is directly proportional to pitch diameter, so you can also compute the ratio from diameters. As Encyclopaedia Britannica notes, the ratio is fixed by the tooth counts of the meshing pair, which is why gears deliver precise, non-slipping speed and torque conversion.

Output speed is inversely proportional to gear ratio: ω₂ = ω₁ / GR, where ω₁ is input speed and ω₂ is output speed.

A 3:1 reduction turns 1000 RPM input into 1000 / 3 ≈ 333.3 RPM at the output. A ratio below 1 (an overdrive) increases speed instead: a 0.75:1 ratio turns 1000 RPM into about 1333 RPM.

This inverse relationship follows from conservation of the number of teeth passing the mesh point per unit time — the smaller gear must spin faster to keep pace. HyperPhysics (Georgia State University) frames this as the rotational analog of a simple machine trading speed for force.

Torque scales directly with gear ratio: T₂ = T₁ × GR (before losses). A 4:1 reduction increases 10 N·m of input torque to roughly 40 N·m at the output.

This is why trucks and winches use high reduction ratios — they trade rotational speed for pulling force. The trade-off is exact in an ideal system because mechanical power (P = T × ω) is conserved: as torque rises by the ratio, speed falls by the same factor.

Real gears lose about 1–2% of power per mesh to friction, so multiply output torque by efficiency (e.g. 0.98). Torque is measured in newton-metres (N·m), per NIST SI conventions.

For a compound gear train, multiply the ratios of each stage together: GR_total = GR₁ × GR₂ × … × GRₙ. If stage one is 3:1 (45/15) and stage two is 2:1 (40/20), the overall ratio is 3 × 2 = 6:1.

Compounding lets small gears achieve very large reductions without a single oversized gear — a two-stage 6:1 box is far more compact than one 6:1 pair.

Output speed becomes input RPM ÷ total ratio, and output torque becomes input torque × total ratio × combined efficiency. Because each mesh loses a few percent, total efficiency is the product of stage efficiencies: 0.98 × 0.98 ≈ 0.96 for two stages.

Gear ratio itself is dimensionless — a pure number such as 3, often written 3:1.

The physical quantities around it use SI units defined by the International System (BIPM/NIST):

• rotational speed in radians per second (rad/s), though engineering practice commonly uses revolutions per minute (RPM), where 1 RPM = 2π/60 ≈ 0.1047 rad/s
• torque is measured in newton-metres (N·m)
• power in watts (W), with P = T·ω when torque is in N·m and ω is in rad/s

Angular frequency relates to rotational frequency by ω = 2πf. Keeping units consistent — converting RPM to rad/s before combining with torque — prevents the most common power-calculation errors.

To hit a target ratio, choose a minimum tooth count for the smaller gear (typically 12–17 teeth to avoid undercutting) and multiply by the ratio to size the larger gear.

For a 3:1 ratio with a 12-tooth pinion, the driven gear needs 12 × 3 = 36 teeth. If the exact number isn't practical, pick nearby whole numbers and accept a small ratio error, or split the reduction across two stages.

Avoiding a common factor between the two tooth counts (a 'hunting tooth') spreads wear evenly across all teeth. Standards bodies such as AGMA and ISO recommend minimum tooth counts to prevent interference on standard 20° pressure-angle involute gears.

Gear ratios appear anywhere rotation is converted for a task.

• Automotive final drives usually run 2.5:1 to 4.5:1, balancing acceleration against highway RPM.
• Bicycles use chainring-to-sprocket ratios so riders can spin an efficient cadence across terrain.
• Robotics and 3D printers rely on precise reduction gearboxes to convert fast, low-torque motors into slow, high-torque joint motion.
• Wind-turbine gearboxes step slow blade rotation up to generator speed, while industrial conveyors and winches use worm drives for 40:1–100:1 single-stage reductions.

IEEE motor-drive literature and Khan Academy's engineering modules both emphasize matching gear ratio to load so the motor operates near its efficient speed-torque band.

Several errors show up again and again:

• The most frequent error is inverting the formula — dividing driving by driven instead of driven by driving — which flips reduction and overdrive.
• Another is confusing gear ratio with speed ratio; they are reciprocals of the rotational-speed relationship, so keep track of which shaft is input.
• People also forget efficiency losses, treating a 4:1 box as delivering exactly 4× torque when friction removes a few percent per mesh.
• In compound trains, adding stage ratios instead of multiplying them is a classic slip.
• Finally, mixing units — combining RPM directly with N·m torque without converting RPM to rad/s — produces wrong power figures.

Verify with the power check: input power should equal output power divided by efficiency.

Yes — belts, chains, and gears all follow the same ratio principle, just with different measurable inputs.

• For gears and chain sprockets, count teeth: ratio = driven teeth ÷ driving teeth.
• For flat or V-belt pulleys, use pitch diameters: ratio = driven diameter ÷ driving diameter, since the belt speed is common to both wheels.
• Timing (synchronous) belts use teeth counts like sprockets.

The key difference is slip: rigid gears and toothed belts maintain an exact ratio, whereas friction belts can slip slightly under load. Chains lose roughly 3–5% to friction versus about 1–2% per gear mesh, so factor efficiency into torque and power results accordingly.