Calculation
i = z₂ / z₁
More teeth on the driven gear reduce speed and ideally increase torque.
i = z₂/z₁
Gear Ratio Calculator from Tooth Counts
More teeth on the driven gear reduce speed and ideally increase torque.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Gear ratio i — The gear ratio – the relationship between input and output speed (equivalently, driven to driving tooth count). i > 1 means a reduction (output turns slower, torque ideally rises); i < 1 means a speed-up.
- Pinion teeth z₁ — The tooth count of the driving gear (pinion). Source: count them on the gear, or take it from a drawing/parts list.
- Gear teeth z₂ — The tooth count of the driven gear (wheel). Source: count them on the gear, or take it from a drawing.
Example
20/50 teeth give i = 2.5.
Assumptions and limits
Ideal external gearing without slip or losses.
Understand Gear Ratio Calculator from Tooth Counts
This calculator finds the missing quantity among gear ratio i and the tooth counts z₁ (pinion) and z₂ (gear) from the other two. It explains how tooth count determines the speed and torque of a gear pair — the same principle behind a bicycle's gears.
What does this quantity describe?
The ratio of a simple spur gear pair is the ratio of the driven gear's tooth count z₂ to the driving pinion's tooth count z₁: i = z₂/z₁. At i > 1 (a reduction), output speed falls relative to input speed, and torque rises by the same ratio if losses are neglected.
On a bicycle, a small front chainring paired with a large rear sprocket gives a large ratio: the pedals turn several times for each slower turn of the rear wheel, but with noticeably more force — ideal for climbing. A large chainring with a small sprocket reverses this, trading torque for higher speed.
Formula and variables
i = z₂ / z₁
i = z₂ / z₁z₂ = i · z₁z₁ = z₂ / i
| Symbol / input | Meaning |
|---|---|
| Gear ratio i | The gear ratio – the relationship between input and output speed (equivalently, driven to driving tooth count). i > 1 means a reduction (output turns slower, torque ideally rises); i < 1 means a speed-up. |
| Pinion teeth z₁ | The tooth count of the driving gear (pinion). Source: count them on the gear, or take it from a drawing/parts list. |
| Gear teeth z₂ | The tooth count of the driven gear (wheel). Source: count them on the gear, or take it from a drawing. |
Choose the inputs correctly
Enter two of the three quantities. Gear ratio requires both tooth counts, z₂ requires ratio and z₁, z₁ requires ratio and z₂. Tooth counts are always positive integers; the calculator accepts intermediate values for computation, but only integer tooth counts are physically meaningful.
How to use the calculator
Select the target quantity and enter the two known values. For multi-stage gearboxes, make sure you use the tooth counts of the specific gear pair considered, not the whole gearbox — overall ratio is the product of the individual stage ratios.
Worked example
Given z₁ = 20 and z₂ = 50, substitution gives i = 50/20 = 2.5. Output speed therefore falls to 1/2.5 of input speed, while ideal output torque rises to 2.5 times input torque.
Understand the result and units
A ratio of i = 2.5 means the pinion turns 2.5 times for every single turn of the larger gear. For sizing, this means: at constant input power, the output delivers more torque but at a lower speed.
Gear ratio i is a dimensionless ratio. Tooth counts z₁ and z₂ are also dimensionless and given as whole numbers.
Typical applications
Gear ratios are used to design gearboxes, machine tool drives, vehicle transmissions, and conveying or lifting equipment, matching a drive's speed and torque to the application.
Assumptions, limits and common mistakes
The formula describes the ideal, loss-free ratio of a single external gear pair. Real gearboxes additionally have meshing efficiency (typically 96-99% per stage), backlash and bearing friction, so actual delivered torque falls slightly below the ideal value.
Common mistake: Do not confuse gear ratio with the ratio of shaft diameters or module — only the tooth-count ratio matters. Also, for multi-stage gearboxes, do not combine the first and last gear's tooth counts directly without correctly multiplying through the intermediate stages.
Frequently asked questions
What does a gear ratio greater than 1 mean?
A speed reduction: output speed is lower than input speed, and ideal torque is correspondingly higher.
How do I calculate the overall ratio of a multi-stage gearbox?
As the product of each individual gear stage's ratio, not their sum.
Why does torque rise for a speed reduction?
Because mechanical power P = M·ω is conserved in an ideally loss-free transmission: if angular velocity ω falls, torque M must rise by the same ratio.