Gear Ratio Calculator
Calculate the gear ratio between two meshing gears using the formula Ratio = Driven Teeth / Driving Teeth. Find output speed, torque multiplication, and mechanical advantage for any gear pair. See also our Pulley Calculator and Torque Calculator.
How to Calculate Gear Ratio
A gear ratio describes the relationship between two meshing gears — specifically, how many times the driving gear must rotate for the driven gear to complete one revolution. Gears are one of the oldest and most important mechanical devices, used since ancient times to transmit and modify rotational motion. They allow engineers to trade speed for torque (or vice versa) with high efficiency and precision.
To calculate the gear ratio, divide the number of teeth on the driven gear by the number of teeth on the driving gear: Ratio = N_driven / N_driving. A ratio greater than 1 means speed reduction and torque increase (the output shaft turns slower but with more force). A ratio less than 1 means speed increase and torque reduction. The output speed equals the input speed divided by the gear ratio: Speed_out = Speed_in / Ratio.
Gear ratios obey the fundamental law of gearing: the product of speed and torque is constant (ignoring friction losses). If a gear train reduces speed by a factor of 3, it increases torque by a factor of 3. This is conservation of power: P = τ×ω (power = torque × angular velocity). In practice, gear efficiency is typically 95-99% per mesh, so some power is lost to friction and heat.
Compound gear trains use multiple gear pairs in series to achieve large overall ratios. The total ratio is the product of individual ratios. For example, two stages of 3:1 give an overall ratio of 9:1. This is how transmissions in vehicles achieve a wide range of ratios — from low gear (high torque for starting) to high gear (high speed for cruising). Planetary gear sets can achieve very high ratios in compact packages.
Gear Ratio Formula
Simple gear ratio:
Ratio = N_driven / N_driving
Output speed:
Speed_out = Speed_in / Ratio
Output torque (ideal):
Torque_out = Torque_in × Ratio
Compound gear train:
Total Ratio = Ratio₁ × Ratio₂ × Ratio₃ × ...
Power conservation:
P = τ_in × ω_in = τ_out × ω_out (ideal)
With efficiency η:
Torque_out = Torque_in × Ratio × η
Velocity ratio (using diameters):
Ratio = D_driven / D_driving = r_driven / r_driving
Example Calculation
A 20-tooth driving gear meshes with a 60-tooth driven gear. The input shaft rotates at 1000 RPM:
Given: N_driving = 20 teeth, N_driven = 60 teeth, Speed_in = 1000 RPM
Gear ratio = 60/20 = 3:1
Output speed = 1000/3 = 333.33 RPM
Torque multiplication = 3× (output has 3× more torque)
If input torque = 10 N⋅m:
Output torque (ideal) = 10 × 3 = 30 N⋅m
Output torque (η=0.97) = 10 × 3 × 0.97 = 29.1 N⋅m
Power check: P_in = 10 × (1000×2π/60) = 1047 W
P_out = 30 × (333.33×2π/60) = 1047 W ✓
Gear Ratio Reference Table
| Driving (teeth) | Driven (teeth) | Ratio | Output (RPM at 1000 in) |
|---|---|---|---|
| 10 | 40 | 4:1 | 250 RPM |
| 12 | 48 | 4:1 | 250 RPM |
| 15 | 45 | 3:1 | 333 RPM |
| 20 | 60 | 3:1 | 333 RPM |
| 20 | 40 | 2:1 | 500 RPM |
| 25 | 50 | 2:1 | 500 RPM |
| 30 | 30 | 1:1 | 1000 RPM |
| 40 | 20 | 0.5:1 | 2000 RPM |
| 15 | 75 | 5:1 | 200 RPM |
| 12 | 72 | 6:1 | 167 RPM |
Frequently Asked Questions
What is a gear ratio?
A gear ratio is the relationship between the rotational speeds (or number of teeth) of two meshing gears. It equals the number of teeth on the driven gear divided by the teeth on the driving gear. A 3:1 ratio means the driving gear rotates 3 times for every 1 rotation of the driven gear. Gear ratios allow mechanical systems to trade speed for torque — a higher ratio gives more torque but less speed, while a lower ratio gives more speed but less torque. This is fundamental to transmissions, clocks, and industrial machinery.
How do gear ratios affect torque and speed?
Gear ratios inversely relate speed and torque while conserving power. A 3:1 reduction ratio reduces output speed to 1/3 of input speed but triples the output torque. This is because power (P = torque × angular velocity) must be conserved (minus friction losses). In a car, first gear has a high ratio (e.g., 3.5:1) for maximum torque at low speed (starting and climbing). Top gear has a low ratio (e.g., 0.7:1) for high speed at cruising RPM. The transmission allows the engine to operate in its efficient RPM range across all vehicle speeds.
What is a compound gear train?
A compound gear train uses multiple gear pairs in series, with intermediate shafts carrying both a driven gear from one pair and a driving gear for the next pair. The overall ratio is the product of individual ratios. For example, two stages of 4:1 give 16:1 overall. This allows achieving very high ratios in compact space — a single gear pair rarely exceeds 10:1 due to size constraints, but compound trains can easily reach 100:1 or more. Clock mechanisms use compound trains to convert the fast escapement into slow hour-hand rotation.
What are planetary gears?
Planetary (epicyclic) gear sets consist of a central sun gear, an outer ring gear, and planet gears that mesh with both. By holding different elements fixed, various ratios are achieved in a compact, coaxial package. Planetary gears are used in automatic transmissions, bicycle hub gears, and industrial gearboxes. They can achieve high ratios (up to ~10:1 per stage) with excellent power density. Their coaxial design means input and output shafts are aligned, simplifying mechanical layouts.
What is gear efficiency?
Gear efficiency measures how much input power reaches the output. Spur gears typically achieve 95-99% efficiency per mesh. Helical gears are similar but slightly lower due to axial thrust. Worm gears range from 30-90% depending on the lead angle — they can be self-locking (non-backdrivable) at low efficiency. Bevel gears achieve 95-98%. Losses come from tooth sliding friction, bearing friction, oil churning, and windage. Multi-stage gearboxes multiply individual stage efficiencies: three stages at 97% each give 0.97³ = 91.3% overall.
How do you choose the right gear ratio?
Choosing the right gear ratio requires matching the motor/engine characteristics to the load requirements. Key considerations: (1) Required output speed and torque, (2) Motor's optimal operating speed, (3) Available space and weight constraints, (4) Efficiency requirements, (5) Noise and vibration limits, (6) Cost. The ratio should allow the motor to operate near its peak efficiency point while delivering the required output. For variable-speed applications, multiple ratios (transmission) or variable-ratio drives (CVT) may be needed.
Gear Types and Applications
Different gear types serve different purposes. Spur gears (straight teeth) are simplest and most common for parallel shafts. Helical gears (angled teeth) run smoother and quieter but produce axial thrust. Bevel gears transmit motion between intersecting shafts (typically 90°). Worm gears provide very high ratios in a single stage and can be self-locking. Rack and pinion converts rotational to linear motion. Each type has specific advantages in terms of efficiency, noise, load capacity, and manufacturing cost.
In modern engineering, gear design involves sophisticated analysis of tooth profiles (involute curves), contact stress (Hertzian theory), bending stress, lubrication regimes, and dynamic loads. Computer-aided design and finite element analysis allow optimization of gear geometry for specific applications. Advanced materials (case-hardened steel, ceramics, polymers) and manufacturing processes (grinding, honing, shot peening) enable gears to transmit enormous power in compact packages — a modern automotive transmission handles hundreds of horsepower in a package smaller than a shoebox.