Gear Ratio Calculator Math Explained

This calculator works out the gear ratio between two gears from the number of teeth on each, and then shows the resulting speed and torque relationship. You enter the number of teeth on the driving gear and the driven gear, and you get the ratio, plus the output speed for a given input speed and the torque change. The math is simple division, but the way the ratio is expressed and applied makes all the difference in real mechanical systems.

Formula

The gear ratio is defined as the number of teeth on the driven gear divided by the number of teeth on the driving gear:

Gear ratio = T_driven / T_driving

Where:

• T_driving = number of teeth on the input (driving) gear, unitless
• T_driven = number of teeth on the output (driven) gear, unitless

For speed, the relationship is inverse: Output speed = Input speed × (T_driving / T_driven). For torque, the multiplier is the same as the gear ratio: Torque multiplier = T_driven / T_driving (ignoring friction losses).

Worked Example

Using the default values: driving gear has 20 teeth, driven gear has 40 teeth, input speed is 1000 rpm.

1. Gear ratio = 40 / 20 = 2.00 : 1
2. Output speed = 1000 × (20 / 40) = 1000 × 0.5 = 500 rpm
3. Torque multiplier = 40 / 20 = 2.00×

So the driven gear turns at half the speed but delivers twice the torque (in an ideal frictionless system).

Why Gear Ratios Matter

Gears are used to trade speed for torque, or torque for speed. A low gear ratio (driven gear smaller than driving gear) increases output speed but reduces torque. A high gear ratio (driven gear larger than driving gear) reduces speed but increases torque. This trade-off is fundamental in bicycles, cars, industrial machinery, and clocks. In a car, first gear has a high ratio (e.g., 3.5:1) to multiply engine torque for acceleration, while fifth gear has a low ratio (e.g., 0.8:1) to allow high speed at low engine rpm.

How to Choose a Gear Ratio

Choosing a ratio depends on what you need at the output:

• For maximum acceleration or lifting: choose a ratio greater than 1 (driven gear larger). The torque multiplier helps overcome inertia or gravity.
• For maximum speed: choose a ratio less than 1 (driven gear smaller). The output spins faster than the input, but torque drops.
• For a 1:1 ratio: both gears have the same number of teeth; speed and torque are unchanged (useful for reversing direction or transferring power).

Also consider the physical size: a very high ratio requires a large driven gear or a multi-stage gear train, which may not fit. In practice, single-stage ratios above 10:1 are rare due to size and efficiency losses.

Common Mistakes and Misconceptions

A frequent error is confusing which gear is the input. The ratio is always driven ÷ driving, not the other way around. Another mistake is forgetting that speed and torque are inversely related: a ratio of 2:1 doubles torque but halves speed, not both. Also, the gear ratio does not include gear diameter directly; only the tooth count matters for the ratio, assuming the gears mesh correctly (same module/pitch). Finally, real gears have friction and efficiency losses (typically 2–5% per gear pair), so actual output torque is slightly lower than the ideal calculation.

Real-World Examples

Bicycles offer a clear illustration. A bike with a 50-tooth chainring and a 12-tooth rear cog has a ratio of 12/50 = 0.24, meaning the rear wheel spins 0.24 times per pedal revolution, giving high speed but low pedaling ease. In contrast, a 34-tooth chainring with a 28-tooth cog gives a ratio of 28/34 = 0.82, making climbing easier. In a car transmission, first gear might be 3.5:1 and final drive 4.0:1, giving a total ratio of 14:1 at the wheels. That is why the engine can turn 3000 rpm while the wheels turn only about 214 rpm.

Limits of the Simple Formula

The formula assumes ideal meshing and ignores friction, gear tooth profile, and backlash. It also does not account for gear train efficiency, which drops with each additional mesh. For multi-stage gear trains, the total ratio is the product of each stage’s ratio, not the sum. The formula also assumes the driving gear is the one with torque input; if you reverse the direction of power flow, the ratio inverts. Finally, the formula gives the theoretical ratio; actual speed may differ slightly due to manufacturing tolerances.

Frequently Asked Questions

What is a good gear ratio for speed?

A ratio less than 1:1 (driven gear smaller than driving gear) gives higher output speed. For example, a 20-tooth driving gear and 10-tooth driven gear gives a 0.5:1 ratio, doubling the speed. However, torque is halved, so this is only suitable for low-load applications.

How do I calculate gear ratio from rpm?

You don’t need rpm to calculate the ratio; you need tooth counts. But if you know input and output rpm, the ratio is input rpm ÷ output rpm. For instance, 3000 rpm in and 1500 rpm out gives a ratio of 2:1.

Does gear ratio affect torque or speed?

Both, inversely. A ratio greater than 1 increases torque and decreases speed. A ratio less than 1 increases speed and decreases torque. The product of torque and speed (power) remains constant (minus losses).

What is the difference between gear ratio and gear train ratio?

Gear ratio refers to a single pair of gears. Gear train ratio is the product of all individual ratios in a multi-stage system. For example, two stages of 2:1 each give a total ratio of 4:1.

Can I use the number of teeth even if gears are different sizes?

Yes, as long as the gears mesh (same module or pitch), the ratio depends only on tooth count. Two gears with different diameters but the same tooth count will have a 1:1 ratio.

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in teeth_drive | Teeth on driving gear | teeth | 20 | 1
in teeth_driven | Teeth on driven gear | teeth | 40 | 1
in rpm_in | Input speed | rpm | 1000 | 0
out ratio | Gear ratio (driven:driving) | :1 | teeth_driven / teeth_drive | 2
out rpm_out | Output speed | rpm | rpm_in * (teeth_drive / teeth_driven) | 1
out torque_factor | Torque multiplier | x | teeth_driven / teeth_drive | 2