Start with a question
Why does a bigger gear turn more slowly?
Gears transfer rotation between shafts. A bicycle, a clock, or a small machine can use different tooth counts to change how fast a shaft turns.
Before you start: You only need counting and division. We’ll introduce the parts and the units before you turn anything.
Your goal: Predict a driven gear’s direction and speed from the two tooth counts.
One gear drives the other
The driving gear receives the input motion. Its teeth push the teeth on the driven gear. Here the teeth are on the outside of both wheels, so two touching teeth move together while the wheels turn in opposite directions.
Count teeth passing the meeting point
Each tooth that passes the contact region on one wheel corresponds to one tooth on the other. One turn of an 18-tooth driver passes 18 teeth. An equally sized 18-tooth driven gear makes a whole turn; a 36-tooth driven gear makes half a turn.
Use that count to predict speed
Speed in revolutions per minute, or rpm, counts full turns in one minute. Multiply the input speed by the driving tooth count and divide by the driven tooth count. The result is a speed magnitude; the driven direction is opposite.
Words you’ll see
- Driving gear / input
- The gear whose speed and direction you control: the highlighted wheel on the left.
- Driven gear / output
- The wheel on the right, turned by the driving gear.
- Teeth
- The evenly spaced projections around a gear that engage the other gear.
- rpm
- Revolutions per minute: the number of complete turns made in one minute.
- Tooth-count ratio
- Driven teeth divided by driving teeth. A 36-tooth wheel driven by 18 teeth gives 2:1.
A worked example: 18 teeth drive 36 teeth
With a 12 rpm input, the 36-tooth driven wheel rotates at 12 × 18 ÷ 36 = 6 rpm in the opposite direction. Its tooth-count ratio is 36:18, or 2:1: two input turns produce one output turn.
Choose Step four times to make one input turn. Compare the turn counts. Then try 18 and 54 teeth while keeping the input speed the same.
- Driving gear · input
- 18 teeth · 12 rpm clockwise
- Driven gear · output
- 36 teeth · 6 rpm counterclockwise
- Input turns
- 0.00
- Output turns
- 0.00
Turn counts use + for clockwise and − for counterclockwise, viewed from above.
Driven gear
Input speed
Input direction
Reduced motion is on. Step advances both gears without animation. Changing the driven gear resets the turn counts.
Output speed = input speed × driving teeth ÷ driven teeth.
12 × 18 ÷ 36 = 6 rpm, in the opposite direction.
Model assumptions and source
This model describes two external spur gears on parallel shafts. It assumes rigid, continuously engaged teeth of matching size. It models direction and speed, not friction, load, torque, wear, or tooth deformation. Other gear arrangements can have different direction rules.
References: KHK · gear dimensions and fischertechnik · gear ratios and revolutions. The wheel outlines are original Forma illustrations.