Mechanical calculator
Gear Ratio and Speed Calculator
Count the teeth, follow the direction changes, and see what the gear pair does to speed and theoretical torque before metal starts turning.
Sage’s guided lesson
A gear ratio is a trade: speed goes one way, torque goes the other
The tooth counts tell you the ideal motion relationship. They do not tell you whether the teeth are strong enough, the gears actually mesh, or the gearbox will survive. Sage is here to keep those three ideas from getting thrown into one bucket.
The teeth are doing the counting
Ratio comes from tooth count, not wishful thinking
For a simple pair of matching external gears, the ratio is the driven gear tooth count divided by the driver gear tooth count. A 36 tooth driven gear connected to a 12 tooth driver has a 3:1 reduction. The driven gear makes one revolution while the driver makes three.
When both gears use the same tooth system, pitch diameter grows in proportion to tooth count. That is why counting teeth gives the same ratio as comparing pitch diameters.
Name the gear attached to the power source first. Swapping driver and driven changes a 3:1 reduction into a 1:3 increase.
Now for the arithmetic
Speed falls as the ratio rises
Driven speed equals driver speed multiplied by driver teeth and divided by driven teeth. Ideal output torque changes by the opposite ratio, because an ideal gear pair trades rotational speed for torque while transmitting the same power.
Worked example
Driver = 12 teeth; driven = 36 teeth; driver speed = 900 RPMRatio = 36 ÷ 12 = 3:1Driven RPM = 900 × 12 ÷ 36 = 300 RPM8 units of input torque × 3 = 24 ideal output unitsAt 90% estimated efficiency: 24 × 0.90 = 21.6The efficiency entry is only an estimate. Real losses vary with tooth form, lubrication, bearings, alignment, speed, and load.
I know this is off topic, but this reminds me of a joke I heard when teaching this to one of my interns, What do you call a group of musical whales? An orca-stra. Yeah, I don’t see the connection either, but I think it is cute!
A small gear with a surprisingly specific job
Idlers change direction and spacing, not the simple ratio
Every external gear mesh reverses rotation. With no idler, the driven gear turns opposite the driver. One idler adds another reversal, so the final gear turns the same direction as the driver. Two idlers reverse it again.
The idler tooth count appears in both the numerator and denominator of the train calculation, so it cancels out. It can move shafts farther apart, route motion around an obstacle, or reverse direction without changing the overall ratio.
An idler still needs suitable size and strength. “It does not change the ratio” does not mean “it carries no load.”
The ratio can be correct while the gears are completely incompatible
Module, diametral pitch, and pressure angle must agree
Two spur gears normally need the same module or diametral pitch and the same pressure angle to mesh correctly. Tooth count alone is not enough. A 20 tooth metric module gear is not automatically compatible with a 20 tooth inch pitch gear.
Backlash is the intentional clearance between mating teeth. Too little can cause binding and heat; too much can create lost motion, noise, and impact. Backlash affects feel and accuracy, but it does not change the nominal ratio.
The teeth engage around the pitch circles, which are imaginary working circles inside the outside diameters.
Tiny machines, clocks, and one very old surprise
Gears were calculating long before electronic calculators
The ancient Antikythera mechanism used a remarkable collection of bronze gears to model astronomical cycles. Much later, clockmaking pushed gear cutting toward greater precision because timekeeping punishes irregular motion.
The involute tooth profile became important because it maintains a nearly constant speed ratio even when the center distance changes slightly. That makes manufactured gears more forgiving than tooth shapes that demand perfect placement.
A “hunting tooth” combination uses tooth counts with few common factors, causing a given tooth to meet many different teeth before the contact pattern repeats. That can spread wear and reveal why seemingly harmless tooth count choices matter.
Before you let the teeth argue at full speed
A ratio calculation is not a gearbox design
- Confirm module or diametral pitch, pressure angle, face width, material, hardness, and lubrication.
- Check tooth bending strength, surface contact stress, shaft torque, key or spline capacity, bearing load, and housing stiffness.
- Use pitch line speed and manufacturer limits when running at high RPM.
- Guard exposed gears; teeth do not care whether they catch a workpiece, glove, hair, or finger.
The calculator assumes a simple external gear train. Planetary, worm, bevel, internal, differential, and compound trains need their own geometry and sign conventions.
And that is the lesson properly accounted for
The ratio is counted, the direction is known, and nobody has lost a finger
You now know what tooth count controls, why idlers cancel from the ratio, and why matching pitch matters as much as the arithmetic. That earns cheese fries, preferably served nowhere near an unguarded gear train.
Quick answers
Gear Ratio and Speed Calculator questions
Which tooth count goes on top?
Use driven teeth divided by driver teeth when expressing a reduction ratio. The speed formula uses driver RPM × driver teeth ÷ driven teeth.
Do idler gears change the ratio?
Not in a simple external train. Their tooth counts cancel, although each idler adds another direction reversal and still carries load.
Why does a larger driven gear turn more slowly?
More teeth must pass the mesh for one revolution, so the driven gear needs more driver revolutions to complete its own turn.
Does a 3:1 ratio triple real torque?
It triples ideal torque before losses. Real output is lower because gears, bearings, seals, and lubrication consume power.
Can any two gears with the right tooth counts mesh?
No. They must use compatible tooth size and geometry, including module or diametral pitch and pressure angle.
What is backlash?
Backlash is the clearance between mating tooth flanks. It allows lubrication and manufacturing tolerance but creates some lost motion.
What is a hunting tooth ratio?
It is a tooth count relationship that delays repetition of the same tooth contacts, often distributing wear across more tooth pairs.
Before you use the result
This result describes ideal kinematics. It does not establish gear strength, service life, lubrication, guarding, or machine safety.