A belt length calculator for two or three pulleys
This belt length calculator finds the length of a flat, V or timing belt that runs round two pulleys, either open or crossed, or round three pulleys placed at any positions. It can also work backwards: enter the belt length you have and it returns the centre distance, which is what you need when you are choosing a belt for a fixed frame or setting up a 3D printer or CNC axis.
For timing belts you can enter pulleys as tooth counts and a pitch (GT2 is 2 mm), and the calculator converts to pitch diameters, reports the belt length in teeth, and tabulates the whole-tooth belts around that length with the centre distance each one gives.
Lengths are exact geometry rather than the usual approximation, and the common formula is shown beside the exact value so you can see how close it is. Belt thickness, stretch and manufacturing tolerance are not included, so confirm with the belt maker before ordering.
How to use the Belt Length Calculator
- Choose two pulleys open, two pulleys crossed or three pulleys.
- Enter the pulley diameters (pitch diameters for V-belts), or switch to teeth and enter the belt pitch and tooth counts.
- Enter the centre distance, or choose "Centre distance for a belt length" and enter the belt you have.
- For three pulleys type the x and y position of each centre; the calculator wraps the belt round the outside.
- Read the length, the wrap on the small pulley and the nearest whole-tooth belt.
Belt Length Calculator features
Exact geometry
Tangent lengths and wrap arcs, not only the approximate formula.
Both directions
Length from centre distance, or centre distance from a belt length.
Timing belts
Tooth counts, pitch diameters and a table of whole-tooth belts such as GT2.
Three pulleys
Spans, wrap angles and arc lengths for any triangle of centres.
Belt length formulas
For two pulleys with diameters D and d and centre distance C, the common approximation for an open belt is L ≈ 2C + π(D + d)/2 + (D − d)² ÷ 4C. The exact result is L = 2√(C² − ((D − d)/2)²) + [D(π + 2θ) + d(π − 2θ)] ÷ 2 with θ = asin((D − d) ÷ 2C). For a crossed belt the term (D − d) becomes (D + d) and both pulleys wrap by π + 2φ.
Example: pulleys of 100 mm and 50 mm on 300 mm centres need an open belt of 837.70 mm; the approximation gives 837.70 mm too, because the diameters differ little. With a large size difference and short centres the gap grows. With equal pulleys the formula reduces to 2C + πD. For three pulleys the belt is the sum of three external tangents plus radius times turning angle on each pulley, and the turning angles must add up to 360°.
Practical notes
- Timing belts come only in whole numbers of teeth, so choose the nearest belt and adjust the centre distance with a slotted mount or an idler.
- V-belts are specified by pitch length; use the pitch diameter of the groove, not the outside diameter of the pulley.
- Wrap on the small pulley below about 120° reduces the power a belt can carry.
- If a pulley lies inside the loop, such as a tensioner pressing on the belt’s inside, this outside-wrap model does not apply.
Belt Length Calculator FAQ
How do I calculate belt length for two pulleys?
Use L = 2C + π(D + d)/2 + (D − d)² ÷ 4C for a quick estimate, or the exact formula this calculator uses. 100 mm and 50 mm pulleys on 300 mm centres need 837.7 mm.
How do I calculate belt length for three pulleys?
Add the three straight spans between the pulleys to the arc each pulley wraps. Enter the pulley positions and the calculator does it, checking that the wraps add up to 360°.
How do I choose a GT2 belt?
Set the input to teeth with a 2 mm pitch, enter both pulley tooth counts and the centre distance, then pick the whole-tooth belt from the table. A 60-tooth and a 20-tooth pulley 100 mm apart need about 150 teeth.
What is the difference between open and crossed belts?
An open belt keeps both pulleys turning the same way; a crossed belt reverses the direction and wraps more on both pulleys but wears faster.
Does it include belt thickness?
No. Use the pitch line or pitch diameter of the pulleys, which already accounts for the belt’s neutral axis.
Is this the same as the gear ratio calculator?
No. This page gives length and centre distance; the gear ratio calculator handles speed ratios for belts and gears.