What this calculator does
Find out how fast and how strongly a set of gears will turn, just from counting their teeth. When two gears mesh, their teeth pass each other one for one. So a small gear driving a big one has to turn several times for each turn of the big gear. The output turns slower, but with more turning force (torque). The gear ratio measures that trade between speed and torque.
This calculator works out the ratio from tooth counts for a single pair of gears. It also handles a two-stage compound train, where two gears share a middle shaft. Add an input speed and torque to see what comes out. You get the output speed in rpm (turns per minute) and in radians per second. You also get the output torque and which way it turns. You can add a small power loss for each mesh (each place two gears meet), too.
Where it's used
Why this math matters outside the classroom:
- Robotics clubs. Students gear a 1,200 rpm motor down with a 12-tooth gear driving a 36-tooth gear to get 400 rpm.
- Bicycle gearing. A rider with a 44-tooth chainring and 11-tooth cog sees the wheel turn 4 times per pedal turn.
- Winches. A winch maker uses a 12:1 compound train so a 5 N·m motor gives about 60 N·m at the drum.
- Clock making. A clock's hour train needs a 12:1 ratio so the hour hand turns once while the minute hand turns 12 times.
- Machine design. An engineer includes 98% efficiency per mesh, so two stages pass on about 96% of the ideal torque.
Grade level and standards: Grades 6–7; Common Core State Standards for Mathematics 6.RP.A.3, 6.RP.A.1. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose one gear pair or a two-stage compound train.
- Enter the teeth on the driving (input) gear and the driven gear. For a compound train, enter the second pair too.
- Enter the input speed in rpm and the input torque in any unit.
- If you like, lower the efficiency per mesh from 100% to allow for friction. Then read the ratio, output speed and output torque.
What the inputs mean
- Driving gear teeth (input)
- The driving gear is the one connected to the motor, crank or input shaft.
- Driven gear teeth
- The gear it meshes with.
- Second-stage driving gear teeth
- In a compound train, the second driving gear is fixed to the same shaft as the first driven gear. So the two turn together.
- Efficiency per mesh
- Each meshing pair loses some power to friction. Well-made spur gears (ordinary straight-toothed gears) lose roughly 1 to 2% per mesh. Worm gears can lose much more.
Formula and method
gear ratio = driven teeth ÷ driving teeth compound ratio = (driven₁ ÷ driver₁) × (driven₂ ÷ driver₂) output speed = input speed ÷ ratio output torque = input torque × ratio × efficiency per mesh^(number of meshes)
In the formulas, driven₁ and driver₁ are the tooth counts of the first pair, and driven₂ and driver₂ the second pair. Meshing teeth move past each other at the same rate. So the gear with more teeth turns more slowly, in exact proportion. A ratio of 3 means the output turns once for every three input turns.
Perfect gears pass on all the power they receive, and power is torque times speed. So if speed drops to a third, torque rises three times. Friction takes a share at each mesh. That lowers the torque but not the speed. That's why the efficiency is raised to the power of the number of meshes.
Worked example
A 12-tooth gear on a motor turning at 1,200 rpm with 10 N·m of torque drives a 36-tooth gear. What comes out?
- Gear ratio: 36 ÷ 12 = 3.
- Output speed: 1,200 ÷ 3 = 400 rpm.
- Output torque, with no losses: 10 × 3 = 30 N·m.
Result: A 3:1 reduction: the output turns at 400 rpm with 30 N·m of torque, in the opposite direction to the motor.
Understanding the result
A ratio above 1 is a reduction: slower and stronger. A ratio below 1 is an overdrive: faster and weaker. A ratio of exactly 1 changes only the direction.
Each external mesh (two gears touching on their outsides) reverses the direction of turning. One pair reverses it. A two-stage compound train turns it back to the input's direction.
Assumptions and limitations
- Ordinary external spur or helical (slanted-tooth) gears that mesh correctly. Idler gears between two gears change direction but not the ratio.
- The same efficiency applies to every mesh, and speed is unaffected by friction.
- Planetary, worm and rack-and-pinion gears follow different rules and aren't covered.
- Bicycle and chain drives work the same way with sprocket teeth, but this doesn't model wheel size or road speed.
- Gear strength and tooth stress aren't checked. A ratio that works on paper may need bigger gears to carry the torque.
Frequently asked questions
How do I calculate a gear ratio?
Divide the number of teeth on the driven gear by the number on the driving gear. A 15-tooth gear driving a 45-tooth gear gives 45 ÷ 15 = 3, written 3:1.
Does a higher gear ratio mean more speed?
Here, the ratio is driven teeth over driving teeth. So a higher ratio means a slower output with more torque. Some fields, such as cycling, flip it the other way. So check which way a ratio is written.
What does an idler gear do?
An idler sits between two gears. It reverses the direction of the last gear without changing the overall ratio. That's because its teeth cancel out: driven ÷ idler × idler ÷ driver = driven ÷ driver.
Why use a compound train instead of one big gear?
A 12:1 reduction from one pair would need a gear 12 times the size of the pinion (the small driving gear). Two stages of 3:1 and 4:1 on a shared shaft give the same ratio in far less space.
How this calculator is checked
Method: Kinematics of meshing gears (equal pitch-line speed) and conservation of power: ratio = N_driven ÷ N_driver, ω_out = ω_in ÷ ratio, τ_out = τ_in × ratio × η. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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