Tool 12 · Mechanics
Belts come in fixed tooth counts, so the honest design question isn't "how long a belt do I need?" but "which stock belt can I buy, and exactly where do I mount the motor for it?" This tool answers both, and checks that enough teeth are gripping.
On the motor
Equal teeth = 1:1 transmission
Roughly where you want the motor
Tooth counts and centre distance must be positive, and the centre distance must be large enough for the pulleys to clear each other.
Belt spec
pitch diameter PD = pitch × teeth ÷ π
belt length ≈ 2C + (π/2)(PD₁+PD₂) + (PD₂−PD₁)² ÷ (4C)
belt teeth = length ÷ pitch → rounded to a stock size
exact C rebuilt from the chosen belt
ratio = driven teeth ÷ driver teeth
The order matters. Most people compute a belt length, then discover that belts are sold in fixed tooth counts and their number isn't one of them. This tool does it the way a machine gets built: compute the ideal length from your target centre distance, round to a belt you can actually order, then solve backwards for the exact centre distance that belt wants. That last number is the one to put on your drawing or CAD sketch.
Belts move torque the same way gears do — the ratio is just the tooth counts — so a 20 T driving a 60 T is a 3:1 reduction: one third the speed, three times the torque (minus a few percent of losses), exactly as covered in the Gear Ratio Calculator. What belts add over gears is quiet running, tolerance for imperfect alignment, and the ability to place the motor a long way from the load — which is why they dominate 3D printers, camera sliders and robot arm joints alike.
Building a linear axis with this belt? The Stepper Motor Calculator turns a pulley tooth count directly into steps per millimetre for your firmware.
6 mm GT2 is the hobby default and handles typical printer and light robot loads; 9 mm and 10 mm versions roughly scale the torque capacity for heavier axes. Width is the cheapest upgrade available — if a belt is skipping and the mesh count is already healthy, go wider before going more exotic.
Closed loops are cleaner for rotary transmissions and fixed centre distances. Open-ended belt cut to length and clamped at both ends is the standard for linear axes, since it removes the length-matching problem entirely — and lets you tension by pulling rather than by moving the motor.
Closed-loop belts are stocked in set lengths (200, 280, 300, 400, 610 mm and so on in GT2), so the tooth count above is the ideal, not necessarily a catalogue item. Pick the nearest stock length, type its tooth count into the driven-teeth field's neighbour — or simply re-enter a target centre distance that lands on a size you can buy — and use the exact centre distance the tool returns for that belt. An idler or slotted mount absorbs the rest.
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