Gear Design Calculator
Reviewed by Hamza Gujjar, Algorithm Architect & Software Developer
Last updated: August 19, 2026Two Gears, One Ratio That Governs Everything Downstream
Mesh a small gear against a large one and the relationship between them is fixed the moment the tooth counts are chosen: speed drops, torque rises, and the trade-off between the two is exact rather than approximate. Getting the gear ratio right at the design stage is what keeps a motor's output matched to what the driven load actually needs.
The Formula
Output Speed = Input Speed / Gear Ratio
Output Torque = Input Torque × Gear Ratio × (Efficiency / 100)
Efficiency accounts for friction and mesh losses in the gear train — a value below 100% reduces the torque actually delivered at the output shaft below what a perfectly efficient mesh would provide.
Worked Examples
| Driver Teeth | Driven Teeth | Input Speed | Input Torque | Efficiency | Ratio | Output Speed | Output Torque |
|---|---|---|---|---|---|---|---|
| 20 | 60 | 1,800 rpm | 50 N·m | 95% | 3:1 | 600 rpm | 142.5 N·m |
| 15 | 45 | 3,000 rpm | 20 N·m | 90% | 3:1 | 1,000 rpm | 54.0 N·m |
A step-down in speed always produces a proportional step-up in torque before efficiency losses are applied — the gear ratio that cuts speed to a third multiplies ideal torque by three.
Where This Calculation Matters
- Motor-to-load matching — selecting a gear ratio so a motor's rated speed and torque land in the range a driven machine actually needs.
- Gearbox and gear train design — sizing individual mesh stages in a multi-stage reduction to hit a target overall ratio.
- Efficiency budgeting — estimating how much torque is lost to friction across a gear train so downstream components are sized with margin.
How to Use This Calculator
- Enter Driver Gear Teeth and Driven Gear Teeth.
- Enter Input Speed (RPM) and Input Torque (N.m).
- Enter Efficiency (%, default 100) if the mesh isn't ideal.
- Select Calculate to get the gear ratio, output speed, and output torque.
Related Calculations
For belt-and-pulley reductions instead of gears, see the Pulley Ratio Calculator. To size the shaft carrying the resulting output torque, use the Shaft Diameter Calculator.
Principles of Involute Spur Gear Geometry and Kinematics
A gear design calculator computes the geometric dimensions, tooth profiles, speed reduction ratios, and mechanical torque capacities for mating spur gear pairs. In mechanical power transmission engineering, gear teeth are profiled with mathematical Involute Curves (the locus of a point on a taut string unwrapping from a base circle cylinder), guaranteeing a constant angular velocity transmission ratio (conjugate gear action) without mechanical velocity pulsation.
Metric Module (m) vs. Imperial Diametral Pitch (P)
Gears can only mesh together if they share an identical tooth size standard:
Imperial Diametral Pitch (P): P = Number of Teeth (z) / Pitch Diameter (d [inches])
Conversion Identity: Module (m) × Diametral Pitch (P) = 25.40 mm/inch
Standard Involute Spur Gear Geometric Proportions (20° Pressure Angle)
| Gear Parameter | Standard Formula (Metric Module m) | Description |
|---|---|---|
| Pitch Diameter (d) | d = m × z | The theoretical operating pitch circle diameter where teeth mesh |
| Circular Pitch (p) | p = π × m | Distance measured along the pitch circle from tooth to tooth |
| Addendum (ha) | ha = 1.0 × m | Radial height of tooth above pitch circle |
| Dedendum (hf) | hf = 1.25 × m | Radial depth of tooth root below pitch circle (includes clearance) |
| Outside Diameter (da) | da = d + 2m = m × (z + 2) | Total tip-to-tip blank diameter of the gear |
| Center Distance (a) | a = m × (z1 + z2) / 2 | Shaft center-to-center operating distance |
Gear Ratio, Speed Reduction, and Torque Multiplication
The mechanical gear ratio (i) between a driving pinion gear (1) and driven gear (2) is defined as:
Step-by-Step Worked Calculation Example
Example: Sizing a 3:1 Speed Reduction Spur Gear Drive
Problem: An electric motor rotates at 1,800 RPM delivering 10.0 N·m of torque. It drives an output shaft at 600 RPM (3:1 speed reduction) using Module m = 3.0 mm spur gears with a 20° pressure angle. The driving pinion has z1 = 20 teeth. Calculate: (1) The number of teeth on the driven gear z2; (2) The pitch diameters d1 and d2; (3) The center-to-center shaft distance a; and (4) The output shaft torque Torque2.
Step 1: Calculate driven gear tooth count:
Gear Ratio i = 1,800 RPM / 600 RPM = 3.00
z2 = z1 × i = 20 teeth × 3.00 = 60 teeth
Step 2: Calculate pitch diameters:
Pinion Diameter d1 = m × z1 = 3.0 mm × 20 = 60.0 mm
Gear Diameter d2 = m × z2 = 3.0 mm × 60 = 180.0 mm
Step 3: Calculate shaft center distance:
Center Distance a = (d1 + d2) / 2 = (60.0 + 180.0) / 2 = 120.0 mm
Step 4: Calculate output torque:
Torque2 = Torque1 × i = 10.0 N·m × 3.00 = 30.0 N·m
Conclusion: The gearset requires a 20-tooth pinion and 60-tooth gear spaced at 120 mm center distance, delivering 30 N·m output torque at 600 RPM.
Tooth Undercutting and Minimum Pinion Tooth Limits
To prevent hobbing tool interference and structural undercutting (weakening tooth roots), the minimum number of teeth on a standard 20° full-depth spur pinion is: zmin = 2 / sin²(20°) ≈ 17 teeth.
Epicyclic (Planetary) Gear Train Ratios
In automatic vehicle transmissions, wind turbine gearboxes, and robotics joint actuators, compact high-torque speed reductions are achieved using Epicyclic (Planetary) Gearsets consisting of a central Sun gear (s), multiple Planet gears (p), and an outer internal Ring gear (r).
With the outer ring gear held stationary, the gear ratio from the input Sun gear to the output Planet Carrier arm is computed as:
AGMA Bending and Contact Stress Analysis
To prevent mechanical tooth breakage and surface pitting fatigue, gear designers calculate root bending stress using the Lewis Bending Equation modified by American Gear Manufacturers Association (AGMA) standards:
Where Ft is tangential tooth force, b is face width, m is module, Y is the Lewis tooth form factor, and K terms are dynamic overload and load distribution factors.
Helical Gears and Contact Ratio Enhancement
In high-speed automotive transmissions, straight spur gears produce gear whine acoustic noise. Mechanical engineers specify Helical Gears whose teeth are cut at a helix angle β (typically 15° to 30°). The helix angle introduces an axial contact overlap ratio, ensuring multiple teeth engage simultaneously and progressively, reducing mechanical impact stress and operating with near-silent smooth power transmission.
Gear Backlash and Thermal Expansion Tolerances
To prevent tooth binding under thermal expansion, gears are designed with controlled Backlash — the intentional circumferential clearance between non-working tooth flanks (typically 0.03 to 0.05 times module m).
AGMA Gear Quality Grades
High-precision aerospace gearing requires AGMA Quality Grades 12 to 14, requiring precision ground tooth flanks with micron-level profile tolerances.