Article — Gear Ratio Calculator (Speed and Torque)
Gear ratio calculator — speed and torque
A gear ratio is the number of teeth on the driven gear divided by the number of teeth on the driver gear. A 60-tooth gear driven by a 20-tooth gear gives a 3:1 ratio — the output turns three times slower and produces three times the torque. Output power equals input power minus friction losses, so the speed and torque tradeoff is fixed.
Every mechanical system that converts rotation between two speeds — cars, bicycles, drills, watch movements, wind turbines, robot arms — runs on gear ratios. Knowing the ratio tells you what the output speed will be, how much torque you can develop, and which gear stage to design for the application.
What is a gear ratio?
The gear ratio is a ratio of tooth counts that equals the inverse ratio of rotational speeds. If gear A has 20 teeth and gear B has 60, B has three times the circumference. When A makes a full turn, B has only turned 20/60 = one third of a turn. So the ratio N₂/N₁ = 60/20 = 3 means a 3:1 reduction.
The ratio also dictates torque. Power transfer is conserved, so reducing speed by a factor of three multiplies torque by three. That is why low gears in cars give acceleration — the engine turns fast but the wheels turn slowly with huge torque. Reverse the relationship and you get overdrive: fast wheels, low torque, fuel economy.
The oldest known gear is from the Antikythera mechanism, a Greek astronomical calculator dated to around 100 BC. It used dozens of bronze gears to predict eclipses and planetary positions. Modern engineering recognizes the same principles still — tooth count ratios determining speed and torque transformation.
Gear ratio formula explained
One core equation: GR = N₂/N₁ = ω₁/ω₂. The first form uses tooth counts, the second uses angular speeds. Both produce the same number for an ideal gear pair, which is the consistency check.
GR N₂ / N₁ω₂ ω₁ / GRτ₂ τ₁ × GRCompound GR₁ × GR₂ × …Output torque scales with the ratio. Output speed scales inversely. Multiply them and you recover input power (minus losses). That conservation law is why you can never get free energy from a gear train — only redistribution.
Three types of gear ratio
Gear ratios fall into three buckets based on whether the output speeds up, slows down, or stays the same.
Reduction gears handle starting and pulling — drills, winches, first gear in a car, lowest cog on a bike. Overdrive gears reduce engine wear at cruising speed — fifth and sixth gear, top sprocket on a road bike. Direct drive (1:1) is the cleanest energy transfer and appears in some EV reducers and high-end audio belt drives.
Car transmission gear ratios
A typical six-speed manual car uses ratios decreasing from about 4:1 in first gear to roughly 0.8:1 in sixth. The engine's most efficient operating range is narrow (around 2,000–3,000 rpm for most gasoline engines), so transmissions step the engine speed up and down to keep the engine in its happy zone while the car covers 0–120 mph.
- First gear ≈ 3.5–4.2:1 — get the car moving from rest.
- Second gear ≈ 2.0–2.5:1 — low-speed acceleration.
- Third gear ≈ 1.3–1.7:1 — city speeds.
- Fourth gear ≈ 1.0:1 — direct drive, no torque change.
- Fifth/sixth ≈ 0.7–0.9:1 — highway overdrive, fuel economy.
- Reverse ≈ 3.2–4.0:1 — high torque, low speed for parking.
Bicycle gear ratios
Bicycle gear ratios are chainring teeth divided by rear cog teeth. A 50-tooth chainring driving an 11-tooth cog gives 50/11 = 4.55:1 — the rear wheel turns 4.55 times per pedal stroke. With a 27-inch wheel, that is 27 × π × 4.55 = 386 inches per pedal stroke, or 9.8 meters.
Cyclists use "gear inches" to compare ratios across different wheel sizes. Gear inches = (chainring teeth / cog teeth) × wheel diameter. A road bike's top gear is around 120 gear inches. A mountain bike's granny gear is below 20. Comparable between any bikes.
Compound gear trains
Real gearboxes rarely use a single stage. Stacking gears in series multiplies the ratios. A two-stage reduction with ratios 3:1 and 4:1 gives a total of 12:1 — the input is twelve times faster than the output, and torque is multiplied by twelve.
Wind turbines use multi-stage gearboxes to convert low blade speeds (around 15 rpm) to high generator speeds (around 1500 rpm) — a 100:1 ratio. Industrial planetary gearboxes can achieve 10,000:1 reductions in a compact package, common in robot joints where small motors must produce large forces.
Gear ratio efficiency and losses
Real gears lose 1–3 percent of power per stage to friction, oil churning, and tooth flexing. A two-stage reduction with two pairs of meshing gears might run at 94 percent efficiency overall. Worm gears are notably worse — 50–80 percent efficient — because the sliding contact at the worm interface generates heat.
The "power in equals power out" rule only holds in the ideal case. Friction is real, oil viscosity drag is real, and very high ratios magnify the friction effects. Industrial gearboxes specify their efficiency rating — usually 92–98 percent for well-designed single stages.
Common gear ratio mistakes
Four traps catch students and hobbyist builders. First, swapping which gear is "driver" and which is "driven" — that inverts the ratio and the torque direction. Second, ignoring friction losses when sizing motors. Third, forgetting that intermediate idler gears do not change the overall ratio between input and output — they only reverse rotation direction. Fourth, treating tooth count as the only variable — module (tooth size) must match for gears to mesh properly. Two gears with different module values cannot operate together.
A fifth subtle issue is gear backlash — the tiny clearance built into every gear pair. Backlash prevents binding and allows lubrication, but it introduces play in the system. In CNC machines, robotics, and precision instruments, designers fight backlash with anti-backlash gears, preloaded systems, or harmonic drives. Ignoring backlash in a position-critical application produces unpredictable accuracy and oscillation problems that the ideal gear ratio formula cannot capture.