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Pulley Belt Length Calculator

Calculators · Added

A belt around two pulleys is two straight runs and two arcs, and the closed form for its length has been in drive handbooks for a century. This computes it in either direction — length from a centre distance, or centre distance from a belt you already own — and reports the wrap angle on the small pulley, which is the number that decides whether the drive can transmit anything without slipping.

What do you want to find?
Belt arrangement

The usual arrangement. Both pulleys turn the same way.

mm
mm
mm

Shaft centre to shaft centre.

RPM

Adds driven speed and belt speed to the result.

How to use the pulley belt length calculator

  1. 1Choose whether you are solving for belt length or for centre distance.
  2. 2Enter both pulley diameters in the same unit, measured where the belt actually rides.
  3. 3Enter the centre distance, or the belt length if you are working the other way.
  4. 4Add the driver speed to get the driven speed and belt speed, and check the wrap angle before ordering anything.

Examples

A motor driving a countershaft

Input
80 mm and 200 mm pulleys, 400 mm centres, open belt
Result
1248.8 mm belt, 162.8° wrap on the small pulley

Comfortably above the 120° that rating tables treat as the reference, so no derating is needed.

Fitting a belt you already have

Input
80 mm and 200 mm pulleys, 1250 mm belt
Result
Centres at 400.6 mm

Solved from the quadratic form of the length equation, so it is exact rather than iterated.

Too big a step down

Input
40 mm and 300 mm pulleys, 180 mm centres
Result
87.5° of wrap — the drive will slip under load

The geometry is valid and the drive is not. Move the centres apart or add an idler on the slack side.

About the pulley belt length calculator

Where the length formula comes from

If both pulleys were the same size, the belt would be two straight runs of exactly the centre distance plus two half circles, giving 2C + pi x D. Unequal pulleys tilt the straight runs slightly and change how much of each pulley is wrapped, and the standard handbook expression accounts for that with one correction term: L = 2C + (pi/2)(D + d) + (D - d)^2 / 4C for an open belt.

That correction is an approximation of the exact expression, which involves the arcsine of (D - d)/2C. For any sensible centre distance the difference is far below a millimetre, which is why the approximation has survived. The wrap angles, though, are computed here from the arcsine directly rather than approximated, because they feed into a power rating and deserve to be right.

A crossed belt swaps (D - d) for (D + d) in the correction and wraps both pulleys by more than half a turn, which is why it is longer than an open belt on the same pulleys.

Reading the centre distance backwards

The more common real problem is the reverse: the belt exists, the pulleys exist, and the question is where to put the shafts. Multiplying the length equation through by 4C turns it into a quadratic in C, and the positive root is C = [b + sqrt(b^2 - 32(D - d)^2)] / 16 with b = 4L - 2 pi (D + d).

A negative discriminant there means no centre distance produces that belt length for those pulleys — the belt is simply too short to pass round both — and the tool says so rather than returning a complex number. That case usually means the belt length was quoted as an outside circumference when the pitch length was needed, or that the diameters were measured on the wrong surface.

Where this stops

Everything here is centreline geometry for two pulleys in a plane. It does not cover a serpentine layout with idlers, a drive where the pulleys are on non-parallel shafts, or the tension needed to transmit a given torque without slipping. Belt tension in particular is a measurement made after installation with a tension gauge or by deflection under a known force, not something derivable from a length.

It also says nothing about belt life. Bending around a small pulley is what fatigues a belt, so a drive that is geometrically fine can still wear belts quickly if the small pulley is below the minimum diameter for the belt section. That minimum is on the manufacturer's data sheet, and it is worth checking before the geometry is fixed by drilling the mounting holes.

Frequently asked questions

Do I measure the pulley across the outside?
Not for a V-belt. The belt rides on the sides of the groove and sits partway down it, so the diameter that matters is the pitch diameter where the belt's own neutral axis runs — always smaller than the outside diameter of the sheave. Using the outside figure makes the calculated belt come out too long, and on a small pulley the error is easily several percent. Flat belts and toothed belts are different again: a flat belt rides on the outside, and a timing belt is specified by tooth count and pitch rather than by diameter at all.
Why does the wrap angle matter more than the length?
Because a friction drive transmits torque through the difference in tension between the tight and slack sides, and how large that difference can grow before the belt slips depends on how far the belt wraps around the pulley. The small pulley always has the lesser wrap, so it is the one that limits the drive. Manufacturers' power ratings are quoted at 180 degrees and corrected downwards below that; by about 120 degrees the correction is significant, and by 90 the drive is usually not workable without an idler to increase the contact.
What is the difference between an open and a crossed belt?
An open belt runs directly between the pulleys and both turn the same way. A crossed belt figure-eights between them, so the driven pulley turns backwards, and both pulleys get more than 180 degrees of wrap — which is why crossed drives grip well. The cost is that the belt rubs against itself where the spans cross, so it wears quickly, and the twist rules out V-belts entirely: only a flat belt tolerates it. Crossed drives survive today mostly on slow, lightly loaded machinery.
How much adjustment should I leave in the centre distance?
Enough to fit the belt without levering it over the pulley, and enough to take up the stretch afterwards. Handbook practice is to allow movement towards the other pulley of a couple of percent of the belt length for installation, and away from it of a similar amount for tensioning — a belt that has to be forced on has already been damaged internally, and one with no take-up left will slip within weeks. Levering a belt on over the sheave flange breaks the tensile cords, and the failure appears later as a belt that stretches unevenly.
Does this tell me what belt to buy?
It tells you the length and geometry, not the belt. Choosing the section — A, B, SPZ, a poly-V profile, a timing belt pitch — needs the power being transmitted, the speed, the service factor for the driven machine, and how many hours a day it runs, all of which come from the manufacturer's selection tables. The wrap angle computed here feeds into those tables as the arc-of-contact correction, so this is the input to that process rather than a replacement for it.

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