EngineeringFormulaMechanical Engineering
Gear Ratio Calculator
Calculate ideal gear ratio, driven RPM, or the target driver or driven tooth count for a simple external gear pair.
A simple gear pair trades rotational speed against torque through the ratio of tooth counts. For two meshed external gears with the same tooth pitch, the ideal magnitude relationship is n1 / n2 = N2 / N1, where 1 is the driver and 2 is the driven gear. This calculator uses that relation to solve for gear ratio, driven speed, or the target tooth count needed to reach a desired reduction or speed-up.
N1 · Whole-number tooth count on the driving gear or pinion.
N2 · Whole-number tooth count on the driven gear.
This page uses the ideal single-stage external-gear magnitude relation only. It does not check tooth interference, center distance, backlash, or strength.
Solution
Enter the required values to calculate gear ratio.
i = N2 / N1
Formula Sheet
- iGear Ratio
- n1Driver Speed
- n2Driven Speed
- N1Driver Gear Teeth
- N2Driven Gear Teeth
Variables & Units
| Symbol | Variable | Description | Common Units |
|---|---|---|---|
| i | Gear Ratio | Drive ratio defined here as driver speed divided by driven speed, equal to driven tooth count divided by driver tooth count. | |
| n1 | Driver Speed | Rotational speed of the input or driving gear. | rpm, rev/s, Hz, rad/s |
| n2 | Driven Speed | Rotational speed of the output or driven gear. | rpm, rev/s, Hz, rad/s |
| N1 | Driver Gear Teeth | Whole-number tooth count on the driving gear or pinion. | |
| N2 | Driven Gear Teeth | Whole-number tooth count on the driven gear. |
How to Use This Calculator
- 01Choose what to solve for first: gear ratio, driven speed, driver teeth, or driven teeth.
- 02Enter the known values. Tooth counts must be whole numbers because real gears cannot have fractional teeth.
- 03Use the ratio convention shown on this page: i = n1 / n2 = N2 / N1, where 1 is the driver and 2 is the driven gear.
- 04Select Calculate to see the result, the active formula, and a substitution line in coherent units.
- 05If the calculator returns a fractional tooth count while solving for a gear size target, treat it as a preliminary target only and choose a nearby practical integer tooth count before finalizing the design.
- 06Use the result as a speed/torque tradeoff check: ignoring losses, a 3:1 reduction gives roughly one-third output speed and roughly three times output torque. Real output torque is lower after gearbox efficiency.
How the Formula Works
At the pitch point of two ideal meshed gears, the tangential speed is the same on both gears. Since tangential speed equals angular speed times pitch radius, we get the inverse size-speed relationship n1 r1 = n2 r2. For gears with the same tooth pitch, pitch radius is proportional to tooth count, so the relation becomes n1 / n2 = N2 / N1.
That means a larger driven gear rotates more slowly than the driver, while a smaller driven gear rotates faster. On this page, a ratio greater than 1 means speed reduction at the driven shaft, and a ratio below 1 means speed increase. The direction reversal of a simple external gear pair is real, but this calculator reports ratio magnitude only.
Because tooth count must be an integer, exact target ratios are not always achievable with a single stage. If a back-solved tooth count is fractional, it tells you the mathematical target; a real design must round to feasible integers and then verify the actual resulting ratio, center distance, and tooth geometry constraints.
Worked Example 01
Gear ratio from tooth counts
Known
- Driver Gear Teeth (N1): 20
- Driven Gear Teeth (N2): 60
Formula
i = N2 / N1
Substitution
i = 60 / 20
Result
i = 3
A 60-tooth driven gear meshed with a 20-tooth driver gives a 3:1 reduction ratio, so the driven shaft turns at one-third of the driver speed in the ideal model.
Worked Example 02
Driven speed from driver speed and tooth counts
Known
- Driver Speed (n1): 1,800 rpm
- Driver Gear Teeth (N1): 20
- Driven Gear Teeth (N2): 60
Formula
n2 = n1 N1 / N2
Substitution
n2 = 1,800 × 20 / 60
Result
n2 = 600 rpm
Because the driven gear has three times as many teeth as the driver, the driven shaft rotates three times more slowly in the ideal model.
Worked Example 03
Required driver tooth count
Known
- Gear Ratio (i): 3
- Driven Gear Teeth (N2): 60
Formula
N1 = N2 / i
Substitution
N1 = 60 / 3
Result
N1 = 20 teeth
To achieve a 3:1 reduction with a 60-tooth driven gear, the driver should have 20 teeth.
Worked Example 04
Required driven tooth count for speed increase
Known
- Gear Ratio (i): 0.5
- Driver Gear Teeth (N1): 40
Formula
N2 = i N1
Substitution
N2 = 0.5 × 40
Result
N2 = 20 teeth
A ratio below 1 means speed increase. Using a 20-tooth driven gear with a 40-tooth driver doubles the driven shaft speed in the ideal model.
Applications
- 01Preliminary sizing of a simple reduction or speed-up gear pair
- 02Checking whether a tooth-count change will reduce or increase output RPM
- 03Quick textbook, lab, robotics, drivetrain, and machine-design calculations
Assumptions
- 01Uses the ideal simple-gear relation i = n1 / n2 = N2 / N1 for a single external gear pair.
- 02Both gears are assumed to have compatible tooth pitch, so pitch radius is proportional to tooth count.
- 03The calculator reports ratio magnitude only; it does not track reversal of rotation direction.
Where This Model Stops
- 01Does not model backlash, efficiency, tooth strength, interference, contact ratio, or manufacturing constraints.
- 02Idler gears, compound gear trains, planetary gear sets, and internal gears need additional analysis beyond this single-stage magnitude relation.
- 03When solving for tooth count, a fractional result is only a target value - real gears require integer tooth counts and must be checked again after rounding.
- 04Does not choose pressure angle, module/diametral pitch, face width, material, lubrication, bearing loads, or gearbox service factor.
- 05Multi-stage reductions multiply stage ratios together; do not enter a single pair's tooth counts and assume it represents the whole train unless it is the only active stage.
References
- [1]Belt and Gear Driven Systems
Pennsylvania State University Mechanics Map
Defines gear ratio as input angular speed divided by output angular speed and gives the equivalent tooth-count relation for a simple gear pair.
- [2]Couple Rotational Motion with Gears
MathWorks Simscape Driveline
States that ideal meshed gears share contact-point motion and gives gear ratio as the ratio of radii or tooth counts.
- [3]10.1 Rotational Variables
OpenStax University Physics Volume 1
Provides the tangential-speed relation vt = rω used to justify the inverse size-speed relationship for meshed gears.
Frequently Asked Questions
What gear-ratio convention does this calculator use?
It uses i = n1 / n2 = N2 / N1, where 1 is the driver and 2 is the driven gear. Under that convention, ratios above 1 mean speed reduction and ratios below 1 mean speed increase.
Do idler gears change the ratio?
No. In a simple train, an idler changes direction but not the overall magnitude ratio between the first driver and final driven gear. The ratio magnitude still depends on the effective input and output tooth counts.
Why can a solved tooth count be fractional if real gears need whole teeth?
Because the algebra returns the exact mathematical target ratio. Real gears must use integer tooth counts, so a fractional result tells you the ideal target before rounding to a practical nearby whole number and checking the actual achieved ratio.
Does a higher gear ratio always mean more torque?
In the ideal model, a reduction ratio above 1 multiplies output torque by roughly the same ratio while reducing speed. In a real gearbox, friction and mesh losses reduce the available output torque, so apply an efficiency or service-factor check before selecting hardware.
How is this different from a sprocket or pulley ratio calculator?
The speed-ratio idea is similar, but gears use tooth counts on meshing gears, sprockets use chain pitch/tooth counts, and pulleys usually use pitch diameters and may slip unless the belt is toothed.