Belt Length Calculator

What the belt length calculator does

To connect two pulleys with a belt, you need to know how long that belt has to be. This calculator works it out from the diameters of the large and small pulleys and the distance between their centres. It gives two answers: the exact geometric length, and the simpler engineering approximation that workshops have long used.

Below is how the belt wraps the pulleys, the two formulas behind the answers, and a worked example.

How to use it

1. Enter the diameter of the large pulley and of the small pulley, each with its own unit.
2. Enter the centre distance, the gap between the two pulley shafts.
3. Press Calculate for the exact length and the engineering approximation, or Reset to clear it.

The geometry of a belt around two pulleys

A belt around two pulleys is made of four parts: two straight runs that bridge the gap between the pulleys, and two curved arcs where the belt hugs each pulley. Add up those four lengths and you have the belt. The straight runs are the tangent lines that just touch both pulleys, and the arcs are the portions of each pulley's circumference the belt wraps around.

When the two pulleys are the same size, each is wrapped by exactly half its circumference, and the straight runs are exactly the centre distance. When the pulleys differ, the belt wraps a little more of the small pulley and a little less of the large one, and the straight runs tilt slightly, growing a touch longer. The exact formula accounts for all of this, while the approximation assumes the simple half-and-half wrap.

The two formulas it uses

With D for the large diameter, d for the small diameter, and C for the centre distance, the engineering approximation is the one most workshops use:

L ≈ (π/2)(D + d) + 2C + (D − d)² ÷ (4C)

The first term wraps half of each pulley, the second is the two straight runs taken as the centre distance each way, and the last is a small correction for the size difference. The exact length, accounting for the true wrap angles and the tilt of the straight runs, is:

L = (π/2)(D + d) + (D − d) · arcsin( (D − d) ÷ 2C ) + 2√( C² − ¼(D − d)² )

For most real drives the two agree to within a hair, which is why the simpler one has stood the test of time.

When the approximation holds

The engineering approximation is excellent in the conditions real belt drives are built for: pulleys of fairly similar size, set a good distance apart. There the wrap on each pulley really is close to half its circumference, and the small correction term mops up most of the remaining difference, so the simple formula lands within a fraction of a percent of the exact length.

It drifts only in the awkward case of one pulley far larger than the other with the two crammed close together. Then the wrap angles swing well away from half and half, and the approximation starts to show its seams. Such layouts are rare in practice, since they make for a poor drive, so the approximation serves almost every real job, and the exact figure is here for when you want to be sure.

Units and precision

You can enter the diameters and centre distance in millimetres, centimetres, metres, inches, or feet, and read the belt length back in any of those, with the calculator converting internally. The large pulley must be entered as at least as big as the small one, and the centre distance must exceed half the sum of the diameters, since otherwise the pulleys would overlap. The result carries several figures, more than belt sizing needs, since in practice you round up to the nearest standard belt length.

A worked example

Take a large pulley 0.2 m across, a small pulley 0.1 m across, with their centres 0.5 m apart.

The approximation gives L ≈ (π/2)(0.3) + 2(0.5) + (0.1)² ÷ 2 = 0.471 + 1.000 + 0.005 ≈ 1.476 m. The exact formula gives 1.476 m as well, the two agreeing closely because the pulleys are a sensible size and well spaced. You would then pick the nearest standard belt at or just above this length.

Questions people ask

How do you calculate belt length?

Add the wrap around both pulleys to the two straight runs between them. The common formula is L ≈ (π/2)(D + d) + 2C + (D − d)²/(4C), using the diameters and the centre distance.

What is the difference between the exact and approximate lengths?

The exact formula accounts for the true wrap angles when the pulleys differ in size. The approximation assumes each pulley is wrapped halfway. For normal drives the two agree very closely.

Why must the centre distance be large enough?

If the centre distance is less than half the sum of the diameters, the two pulleys would overlap, which is physically impossible. The calculator checks for this.

Should I use this exact length for my belt?

Use it as the target, then choose the nearest standard belt size at or just above it, and adjust the centre distance slightly if needed to take up the difference.

References

A quick note on where the formulas come from. The exact open-belt length from the wrap geometry, and the long-used engineering approximation, are standard mechanical engineering, set out by Engineers Edge and in references such as Machinery's Handbook. The SI units follow the US National Institute of Standards and Technology.

1. Engineers Edge, Flat Belt Length and Pulley Center Distance Calculation. https://www.engineersedge.com/belt_design/belt_length_pulley_center_dist.htm
2. Machinery's Handbook, Industrial Press, belt length and pulley drives.
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811