Belt Length
Calculate the required belt length for a two-pulley drive from the pulley diameters and center distance. Supports open and crossed belt arrangements in metric or imperial units.
Find the Belt Length for Two Pulleys
Belt Length Formula
Where:
- \(L\) — required belt length (same unit as the inputs)
- \(C\) — center distance between the two pulley axes
- \(D\) — pitch (effective) diameter of the large pulley
- \(d\) — pitch (effective) diameter of the small pulley
Use the minus sign \((D - d)\) for an open belt and the plus sign \((D + d)\) for a crossed belt. The formula is the standard engineering approximation: \(2C\) covers the two straight spans, the arc term wraps the belt around both pulleys, and the squared term corrects for the extra length caused by unequal pulley sizes.
Belt Length – Calculation Example
Example: an open belt connects a 200 mm pulley to a 100 mm pulley whose shafts are 500 mm apart.
\[ L \approx 2(500) + \frac{\pi}{2}(200 + 100) + \frac{(200 - 100)^2}{4 \times 500} \]
\[ L \approx 1000 + 471.24 + 5 = 1476.24 \text{ mm} \]
Result: the required belt length is approximately 1476 mm (1.476 m, or 58.1 in). You can reproduce this by leaving the calculator's default values unchanged and pressing Calculate.
How to use the calculator
- Enter the Large pulley diameter (D) — the pitch (effective) diameter of the bigger pulley — and pick its unit (mm, cm, m, in, or ft).
- Enter the Small pulley diameter (d) with its unit.
- Enter the Center distance (C), the distance between the rotation axes of the two pulleys.
- Choose the Belt arrangement: Open belt if the pulleys rotate the same way, or Crossed belt if the belt is twisted into a figure-eight so the pulleys rotate in opposite directions.
- Read the approximate belt length, shown in your chosen unit plus a table of all common units, along with the geometry details (wrap angles, straight span length, and arc lengths on each pulley).
What the belt length calculation means
A belt drive transfers power between two shafts by friction (flat and V-belts) or by teeth (timing belts). To size or order a belt you need its total length, which is the sum of three parts: the two straight spans running between the pulleys, and the two arcs of contact wrapped around each pulley. Because the pulleys usually have different diameters, the belt leaves each pulley at a slight angle rather than along the line of centers — this is what the \((D \mp d)^2/(4C)\) correction term accounts for.
The geometry details tell you how well the drive will actually work. The angle of contact (wrap angle) is how many degrees of each pulley circumference the belt grips. In an open belt the small pulley always has the smaller wrap; if it falls below roughly 120–130°, the belt may slip under load. A crossed belt gives both pulleys a wrap angle larger than 180°, improving grip at the cost of faster wear where the belt rubs against itself at the crossing point.
The center distance ratio \(C/d\) is a useful design check: most practical two-pulley drives work best with \(C/d\) between about 1.5 and 5. Very short center distances force extreme wrap angles and flex the belt too often around the small pulley, while very long distances make the belt sag and flap. Note that the pulley speed ratio is fixed by the diameters alone, \(n_1/n_2 = D/d\) — the belt length does not change it.
Assumptions, limitations and practical tips
- Approximate formula. The formula used here is the classic engineering approximation. It is accurate to well under 1% for typical layouts (center distance larger than the pulley diameters) and becomes exact when the pulleys are equal in size.
- Pitch diameter, not outside diameter. For V-belts and timing belts, use the pitch (effective) diameter from the manufacturer's data — typically 5–10 mm smaller than the outside diameter for a V-belt pulley. Using the outside diameter will overestimate the length.
- Minimum center distance. Geometrically, an open belt needs \(C > |D - d|/2\) and a crossed belt needs \(C > (D + d)/2\); the calculator warns you if the layout is impossible.
- No sag, thickness, or tension allowance. The result is the neutral-axis length of an ideal belt. Real belts stretch slightly and take-up (adjustable) pulley mounts are recommended so you can set the tension after installation. For a V-belt, choose the nearest standard belt section and length from a manufacturer's catalogue rather than ordering a custom length.
- Buying a joined (endless) belt? Its length is measured under a specified tension and may differ by a fraction of a percent from the calculated geometric length — always check the catalogue tolerance.
If you are designing the wider mechanical or fluid system around the belt drive, you may also find our Pump Power calculator useful for sizing the motor that drives the pulley, and the Reynolds Number and Pipe Pressure Drop calculators for the flow side of the system.
Frequently Asked Questions
How do I measure the pulley diameter correctly?
Measure the pitch (effective) diameter, not the outside edge. For a flat-belt pulley this is roughly the outer surface the belt rides on, but for a V-belt pulley it is the diameter at the level where the belt's neutral axis sits — typically 5–10 mm smaller than the outside diameter. For timing belt pulleys use the pitch diameter printed in the catalogue. Measuring the outside diameter of a V-belt pulley is the single most common source of overestimated belt lengths.
What is the belt length for two equal pulleys?
When both pulleys have the same diameter, the correction term (d - d)² = 0 disappears and the approximate formula becomes exact: L = 2C + πD. For example, two 150 mm pulleys with 400 mm between centers need L = 800 + π × 150 ≈ 1271 mm of belt for an open drive. Each pulley is wrapped exactly 180°, so both spans are parallel to the line of centers.
How do I find the center distance if I already know the belt length?
Rearrange the formula into a quadratic in C. For an open belt, L = 2C + π(D+d)/2 + (D−d)²/(4C) becomes 8C² − (4L − 2π(D+d))C + (D−d)² = 0, and you take the larger root with the quadratic formula. The smaller root corresponds to an unrealistically tight layout. If the result is negative, no center distance works with that belt — pick a shorter belt or larger pulleys.
Why does a crossed belt need a longer belt than an open one?
In a crossed (figure-eight) belt, the belt wraps more than half of each pulley, so both arcs of contact grow and the straight spans run at a steeper angle. Geometrically, the correction term uses (D + d)² instead of (D − d)², which is always larger. The gain in wrap angle (both pulleys exceed 180°) improves grip and allows opposite rotation directions, but the belt rubs against itself at the crossing point and wears out faster.
How accurate is the calculated belt length, and how much tolerance should I allow?
The approximation is accurate to well under 1 % whenever the center distance exceeds the pulley diameters, and it is exact for equal pulleys. In practice, choose an adjustable (take-up) mounting that lets you change the center distance by roughly 1–2 % of the belt length, since real belts stretch slightly and manufacturers' endless belts are measured under specified tension. Never order a custom belt from the raw geometric number — round to the nearest standard catalogue length.
Can I mix units, e.g. pulley diameters in inches and center distance in millimeters?
Yes — the calculator converts each input independently, so you can enter each quantity in its own unit and the result is converted correctly. Just make sure you enter each value in the unit shown next to its field. If you compute by hand, first convert everything to one unit, because mixing inches and millimeters in the same formula is a frequent mistake that throws the result off by a factor of 25.4.
Does the belt length affect the speed ratio of the pulleys?
No. The speed ratio depends only on the pulley diameters: n₁/n₂ = D/d. Changing the center distance or the belt length changes how much belt wraps each pulley (and therefore the grip and wear), but not how fast the driven shaft turns. A longer belt on the same pulleys simply needs a larger center distance or an idler to take up the slack.