T5 Timing Belts, pulleys, and why radians are useful.

To be confusing, I will use tau rather than pi.

The commonly used (RepRap Prusa, etc) T5 timing belt has a pitch of 5mm. This means that the radius of a pulley of N teeth is N*5mm/tau.

Most small NEMA17 steppers will do 1.8 deg/step. If we convert this into radians, this is tau/200 radians per step.

A pulley of radius R subtends R*angle if rotated about angle. What this means is that if our stepper does tau/200 radians/step, and a pulley of radius R is mounted on it, a point on the edge of the pulley will move R*tau/200 every step.

To put it all together:
Teeth = N

Pulley radius = N *5mm/tau

Step angle = tau/200

Step distance = N*5mm/tau * tau/200 = N/40 mm

What this means: a 10 tooth pulley has step resolution of 0.25mm.