Gear Calculations for Design: Ratios, Torque and Speed

Gear calculations for design determine how a gear pair changes rotational speed and torque, and help establish tooth counts, center distance, and basic geometry. Enter the driver and driven gear tooth counts plus the input speed and torque to calculate the ratio, output speed, ideal output torque, and center distance for standard spur gears.

Formula

For two external spur gears, the tooth-count ratio determines their speed and ideal torque relationship:

• Ratio: i = z₂ / z₁
• Driven speed: n₂ = n₁ / i
• Ideal driven torque: T₂ = T₁ × i
• Pitch diameter: d = m × z
• Center distance: a = (d₁ + d₂) / 2 = m × (z₁ + z₂) / 2

Here, z₁ and z₂ are the driver and driven tooth counts (teeth); n₁ and n₂ are their rotational speeds in revolutions per minute (rpm); T₁ and T₂ are torque in newton-metres (N·m); m is module in millimetres (mm); d₁ and d₂ are pitch diameters in millimetres; and a is the distance between shaft centers in millimetres. The ratio shown is driven teeth divided by driver teeth, so a value above 1 means speed reduction.

The torque result assumes an ideal, lossless gear pair. Real output torque is lower because of friction, lubrication, bearing drag, and other losses. The calculator uses tooth counts for the ratio and assumes standard compatible spur gears for its center-distance calculation.

Worked Example

With the defaults, the driver has 20 teeth, the driven gear has 40 teeth, input speed is 1,200 rpm, input torque is 10 N·m, and module is 2 mm.

1. Ratio: 40 / 20 = 2.
2. Output speed: 1,200 / 2 = 600 rpm.
3. Ideal output torque: 10 × 2 = 20.00 N·m.
4. Center distance: 2 × (20 + 40) / 2 = 60.00 mm.

Thus, this pair halves rotational speed and ideally doubles torque. The two external gears turn in opposite directions.

How Gear Ratios Work

When two meshing gears have different tooth counts, the smaller gear must rotate more times to move the larger gear through one complete revolution. A 20-tooth driver turning a 40-tooth driven gear therefore produces a 2:1 reduction. Reversing which gear drives the other changes the ratio: a 40-tooth driver turning a 20-tooth gear gives a ratio of 0.5 by this calculator’s convention and doubles output speed rather than reducing it.

For an ideal system, mechanical power is conserved: power is torque multiplied by angular speed. Increasing torque through a reduction corresponds to lower output speed. Gears do not create power; losses make actual output power lower than input power. For a single external mesh, rotation direction reverses. Internal gears and gear trains have different direction behavior.

Choosing Teeth, Module, and Gear Size

Tooth counts set the ratio, but a design must also fit its space and carry the load. Module is a metric measure of tooth size: pitch diameter equals module multiplied by tooth count. For example, a 20-tooth gear with module 2 has a 40 mm pitch diameter. The imperial counterpart commonly used in US designs is diametral pitch (DP), defined as teeth per inch of pitch diameter; a larger DP means smaller teeth. Module and DP are related by m (mm) = 25.4 / DP.

A pair of standard spur gears must use compatible tooth geometry, including the same module (or the same DP), pressure angle, and tooth form. Tooth counts alone do not guarantee that two gears mesh. The calculator’s center distance assumes standard, unshifted gears; intentional profile shifts, backlash adjustments, or special tooth geometry can change the required setting.

• Choose the ratio from required input and output speeds, then select integer tooth counts that approximate it.
• Check the package envelope using pitch and outside diameters; outside diameter is not the same as pitch diameter.
• Check tooth strength, shaft and bearing loads, material, lubrication, operating temperature, and expected duty cycle using appropriate design data.
• Confirm backlash, alignment, manufacturing tolerances, and assembly clearances for the actual application.

Limits, Units, and Common Mistakes

This calculation is a first-pass kinematic and geometry estimate, not a complete gear design or strength rating. It does not determine whether teeth will survive bending or contact stress, whether shafts and bearings are adequate, or how much noise and heat the assembly will produce. Those checks depend on load, speed, materials, tooth geometry, lubrication, service conditions, and applicable engineering methods or standards.

Common errors include reversing the ratio, mixing units, and treating ideal torque as guaranteed output. If a ratio is defined as driver teeth divided by driven teeth instead, the formula for speed must be adjusted consistently. Do not combine millimetres with inches in a diameter or center-distance calculation. In US customary design, use inches with DP: pitch diameter in inches equals tooth count divided by DP, and center distance is half the sum of pitch diameters. Convert torque carefully as needed: 1 N·m is approximately 0.7376 lbf·ft.

Rounding tooth counts can alter the achieved ratio, so use actual integer tooth counts in final calculations. For multiple gear stages, calculate each stage separately; the overall speed ratio is the product of stage ratios, while direction depends on the number and type of meshes. Planetary, bevel, worm, helical, and rack-and-pinion systems require geometry and relationships beyond this simple external spur-gear model.

Frequently Asked Questions

How do I calculate a gear ratio from tooth counts?
Divide the driven gear’s tooth count by the driver’s tooth count when reporting driven teeth per driver tooth. For example, 40 driven teeth divided by 20 driver teeth gives a 2:1 reduction.

How do I calculate output speed?
For a simple pair, divide input speed by the driven-to-driver tooth-count ratio. A 1,200 rpm input with a 2:1 reduction gives 600 rpm, before considering slip or other system effects.

Does a gear reduction increase torque?
Ideally, torque increases by the reduction ratio while speed falls by the same factor. Actual torque gain is smaller because gear meshes and supporting parts have losses.

What is the center distance for two spur gears?
For standard external spur gears, it is half the sum of the pitch diameters. With a shared module, it is module multiplied by the sum of tooth counts, then divided by two.

Can gears with different modules mesh?
Standard gears generally need matching module, pressure angle, and compatible tooth geometry to mesh correctly. Different tooth counts are fine; incompatible tooth sizing or form is not.

Related

• Bike Gear Ratio Speed and Cadence Calculator
• Bicycle Gearing Ratio Calculator and Guide
• Bike Gear Calculator Comparison: Ratios, Speed and Cadence

in z1 | Driver teeth | - | 20 | 0
in z2 | Driven teeth | - | 40 | 0
in n1 | Input speed | rpm | 1200 | 0
in t1 | Input torque | N·m | 10 | 0
in m | Module | mm | 2 | 0
out ratio | Gear ratio (driven/driver) | - | z2 / z1 | 2
out n2 | Output speed (ideal) | rpm | n1 / ratio | 0
out t2 | Output torque (ideal) | N·m | t1 * ratio | 2
out c | Center distance | mm | m * (z1 + z2) / 2 | 2