Gear Ratio Equation Calculator
- Output speed
- RPM
- Result
- Torque multiplier
- x
- Result
- Driver gear teeth
- 20 teeth
- You entered
This gear ratio calculator works out the ratio between two gears from the number of teeth on each, and it also gives you the output speed and torque change for a given input speed. You enter the tooth counts (and optionally the input RPM), and you get the gear ratio, output speed, and torque multiplier. The core equation is simple: gear ratio = driven gear teeth ÷ driver gear teeth.
Formula
The gear ratio R is defined as:
R = Ndriven / Ndriver
Where:
- Ndriver = number of teeth on the input (driving) gear – no unit
- Ndriven = number of teeth on the output (driven) gear – no unit
For a pair of meshed gears, the ratio is purely a tooth count ratio. If the driver has 20 teeth and the driven has 40, the ratio is 40/20 = 2.0, often written 2:1. This means the driven gear turns once for every two turns of the driver.
From the ratio you can derive two useful quantities:
- Output speed = input speed ÷ R (in the same RPM units)
- Torque multiplier = R (assuming 100% efficiency, which real gears never achieve)
Worked Example
Using the default values from the calculator above:
- Driver teeth = 20, driven teeth = 40
- Gear ratio R = 40 / 20 = 2.0 (written 2:1)
- Input speed = 1000 RPM
- Output speed = 1000 / 2.0 = 500 RPM
- Torque multiplier = 2.0 (ideal)
So a 20-tooth gear driving a 40-tooth gear halves the speed and doubles the torque (in an ideal frictionless system). The calculator shows the same numbers: ratio 2.00, output 500 RPM, torque multiplier 2.00.
How Gear Ratios Work
When two gears mesh, their pitch circles roll together without slipping. The number of teeth is proportional to the pitch circle diameter, so the ratio of teeth equals the ratio of diameters. A larger driven gear means the driver must rotate more times to complete one output revolution – that reduces speed but increases torque. Conversely, a smaller driven gear gives higher output speed but lower torque.
This trade-off is the fundamental purpose of gears: you cannot get both more speed and more torque from the same input power. The product of torque and angular speed (power) stays roughly constant, minus friction losses. Real gear trains lose 1–3% per mesh for typical spur gears, more for worm gears.
How to Choose Gear Ratios
Choosing a gear ratio depends on what you are trying to achieve:
- Speed increase: driven gear smaller than driver (R < 1)
- Torque increase: driven gear larger than driver (R > 1)
- Matching a motor to a load: you want the motor to operate near its optimal RPM range while the output runs at the required speed.
- Compound gear trains: multiply the ratios of each meshing pair. For two pairs, total ratio = R1 × R2.
For example, a robot arm motor spinning at 6000 RPM might need 30 RPM at the joint. That requires a 200:1 reduction, which is impractical with a single pair – you would use several stages (e.g., 10:1 × 20:1).
Common Mistakes and Misconceptions
The most frequent error is reversing the ratio – putting the driven teeth in the numerator when you mean the driver. Always ask: “which gear is input?� If the driver has more teeth than the driven, the output is faster, and the ratio is less than 1.
Another mistake is confusing gear ratio with gear train value. A single pair has a ratio; a multi-stage train has a total ratio that is the product of each pair. Some people also forget that the ratio is dimensionless – it is not a speed or torque, just a multiplier.
Finally, the torque multiplier is only true for an ideal gearbox. Real gears have friction, so output torque is slightly lower. For most design work, assume 90–99% efficiency per mesh.
Gear Ratio vs. Speed and Torque
Understanding the relationship between ratio, speed, and torque is critical. The input power is:
Power (W) = Torque (N·m) × Angular speed (rad/s)
Since power is conserved (minus losses), if you double the torque, you halve the speed. The gear ratio directly sets that trade-off. For example, a 10:1 reduction gives 10 times the torque but one-tenth the speed. This is why car transmissions use low gears (high ratio) for acceleration and high gears (low ratio) for cruising.
Limitations of the Simple Equation
The basic gear ratio equation assumes:
- Gears are perfectly rigid and mesh without backlash
- No friction, wear, or heat loss
- Constant input speed
- Gears are of the same type (spur, helical, etc.)
In reality, gear trains have efficiency losses, backlash (play between teeth), and inertia effects. For high-precision applications like CNC machines or robotics, you need to account for these. But for most hobby and educational projects, the simple ratio is accurate enough.
Frequently Asked Questions
What is the gear ratio if the driver has 12 teeth and the driven has 36?
Ratio = 36/12 = 3.0, or 3:1. The driven gear turns three times slower than the driver, but with three times the torque.
How do I calculate the output RPM from a gear ratio?
Divide the input RPM by the gear ratio. For example, 1500 RPM input with a 2.5:1 ratio gives 1500 / 2.5 = 600 RPM output.
Does the gear ratio affect horsepower?
No, horsepower (power) is unchanged by gearing (ignoring losses). Gears trade speed for torque, but the product (power) stays the same. A 10 HP motor delivers 10 HP at the output, just at different speed/torque combinations.
What is the difference between gear ratio and gear train value?
Gear ratio usually refers to a single pair of meshing gears. Gear train value is the total ratio of a multi-stage system, which is the product of all individual pair ratios. For example, two 2:1 pairs give a 4:1 total.
Can I use this calculator for belt or chain drives?
Yes, the same ratio equation applies if you use the number of teeth on the sprockets or pulleys instead of gears. For friction belt drives, you can use the pulley diameters instead of tooth counts.