Gears

How Spur Gears Work: Teeth, Pitch Circles and the Grammar of Rotation

· by Mechanism Lab

Spur gears are the oldest and simplest gears we have. Two cylinders with teeth cut around them, meshing tooth-for-tooth: that is the whole idea. Yet inside that idea lives almost everything you need to understand about every gear that followed — planetary sets, worm drives, even the harmonic drives inside robot joints.

The core principle: teeth as a chain of pushes

When two spur gears mesh, one tooth of the driving gear slides into the gap between two teeth of the driven gear and pushes it. As the gears rotate, the next tooth takes over the push, then the next. Rotation is transferred as a continuous sequence of contact forces at the tooth surfaces.

centre distance = r1 + r2 driver · 12 teeth driven · 24 teeth ratio 2:1 · counter-rotation dashed line = pitch circle
Two spur gears in mesh. The dashed circles are the pitch circles — the imaginary friction rollers that roll without slipping. The 12-tooth driver spins twice for every turn of the 24-tooth driven gear, and the two always rotate in opposite directions.

Watch the animation above: the driver (blue) turns continuously, and the driven gear (grey) turns the opposite way. Two rules are visible immediately:

  1. External spur gears always counter-rotate. If you need the same direction, you add an idler gear between them.
  2. The smaller gear must spin faster. Both gears trace the same distance at their pitch circles, so the one with the shorter circumference compensates with more rotations.

The pitch circle: where the gears “really” touch

There is a circle on each gear — drawn as a dashed line in the diagram — where the two cylinders would behave as if they were friction rollers pressing against each other. This is the pitch circle, and its diameter is the single most important number of a gear.

Gear designers pretend the two pitch circles roll against each other without slipping. All the arithmetic of gears follows from that fiction:

Gear ratio: counting teeth

The gear ratio between two meshing spur gears is simply the ratio of their tooth counts — driven teeth divided by driver teeth.

gear ratio i = Ndriven / Ndriver

A 40-tooth gear driven by a 10-tooth pinion has a ratio of 4:1. The pinion turns four times for every single turn of the big gear. Two consequences follow, and they are the two great trades of machine design:

You gainYou pay
4× the torque at the output¼ of the output speed
¼ of the output speedExtra friction stages if you chain many gears

Why involute teeth?

If gears only needed to push, any tooth shape would do. But machines need smooth push. Modern gears use the involute curve — the path traced by the end of a taut string unwinding from a circle — because it has a remarkable property: as two involute teeth slide through contact, the direction of the pushing force stays constant in space, and the velocity ratio stays exactly constant even as the contact point travels.

That constancy is why a well-made gearbox hums instead of jerking. Cheap or worn gears with incorrect profiles make the ratio wobble slightly four times per revolution — you can feel it as vibration, and hear it as gear whine.

Pressure angle and tooth shape

The most common involute gears share a 20° pressure angle, the angle of the force line between meshing teeth. A higher pressure angle makes teeth stronger but noisier; a lower one runs quieter but transmits less force for the same size. Most industrial spur gears you will ever handle are 20°, full-depth teeth.

Where spur gears lose

Spur teeth engage along their entire width at once. That full-width contact is efficient but loud at high speed, and it limits how fast the gear can spin quietly. That limitation is the reason helical gears exist — and that is a story for another article.

Key takeaways