Belt & Pulley Calculator
Belt length, wrap angle, speed ratio and the tension a drive can hold before it slips — from the exact geometry.
from the geometry, not the usual approximation
ratio 2 — a step-up, the driven pulley turns faster
π × driver diameter × rev/s
| Straight span (each side) | 896.87 mm | √(C² − ((D − d) ÷ 2)²) — the belt leaves each pulley on a common tangent |
| Wrap on the large pulley | 189.56° | always 180° or more on an open belt |
| Wrap on the small pulley | 170.44° | the two wrap angles always add to 360° |
| Tension ratio before slip | 2.441 | e^(µθ) on the smaller wrap — the most the tight side can exceed the slack side |
| Textbook approximation | 2513.11 mm | 2C + π(D + d) ÷ 2 + (D − d)² ÷ 4C — off by 0.004 mm here (0.0001%) |
A belt runs along a common tangent to both pulleys and wraps each through a definite angle, so its length is 2C·cos α + R·θ_R + r·θ_r — two straight spans plus two arcs. Speed ratio is simply the diameter ratio: N₂ = N₁ × D₁ ÷ D₂.
The length is geometry, not a rule of thumb
Almost every handbook prints belt length as an approximation: L ≈ 2C + π(D + d) ÷ 2 + (D − d)² ÷ 4C. It is a truncated series, and it is very good for ordinary proportions — on a 300 mm and 150 mm pair at 900 mm centres it is wrong by four thousandths of a millimetre. But it is not the definition, and it drifts exactly where a designer is most likely to be pushing: bring those same pulleys to 120 mm centres and the shortcut is 6.35 mm short, which is 0.66% and more than enough to matter when you are choosing a standard belt.
The exact form comes straight from the picture. Let α be the angle the straight span makes with the line of centres, so sin α = (D − d) ÷ 2C for an open belt. Each straight span is then C·cos α, the large pulley is wrapped through π + 2α and the small one through π − 2α, and the four pieces add. When the pulleys are equal, α is zero and the whole thing collapses to L = 2C + πD, exactly as it should.
Wrap angle is what decides whether it slips
The two wrap angles on an open belt always add to 360°: whatever the large pulley gains, the small one loses. That matters because slip starts wherever the wrap is smallest, and the most the tight side can exceed the slack side before it does is given by the capstan equation, T₁ ÷ T₂ = e^(µθ) with θ in radians. Since θ appears in an exponent, losing wrap costs you far more than it looks. A 300/150 drive at 900 mm centres wraps the small pulley through 170°; push the ratio to 400/60 at 500 mm and the small pulley gets only 140°, and the tension the drive can transmit falls with it.
open belt — two straight spans plus the two wrapped arcs; equal pulleys give L = 2C + πD exactly
- 1 Fix the speed ratio from the diameters. N₂ = N₁ × D₁ ÷ D₂. Belt thickness shifts this slightly in reality, because the belt bends about its own pitch line rather than the pulley surface.
- 2 Choose a centre distance. Roughly 1.5 to 2 times the sum of the diameters is the usual starting range — close enough for a compact drive, far enough to keep wrap on the small pulley.
- 3 Compute the belt length, then round to a stock size. Belts come in standard lengths. Pick the nearest available and work the centre distance back from it.
- 4 Check the wrap on the small pulley. Below about 120° the drive is losing most of its grip. Increase the centre distance, or add an idler to wrap it further.
- 5 Check the belt speed. v = π × D × N ÷ 60. Ordinary V-belts are happiest well under about 30 m/s; above that centrifugal force starts lifting the belt out of the groove.
How far the textbook approximation drifts
Open belt, exact geometry against L ≈ 2C + π(D + d) ÷ 2 + (D − d)² ÷ 4C. All lengths in mm.
| Drive | D | d | C | Exact L | Approximate L | Error |
|---|---|---|---|---|---|---|
| Typical V-belt | 300 | 150 | 900 | 2513.11 | 2513.11 | 0.004 mm (0.0001%) |
| Big reduction | 400 | 60 | 500 | 1780.94 | 1780.37 | 0.577 mm (0.032%) |
| Close centres | 300 | 100 | 120 | 958.01 | 951.65 | 6.354 mm (0.663%) |
| Very close | 200 | 80 | 70 | 635.50 | 631.25 | 4.246 mm (0.668%) |
| Long span | 250 | 120 | 3000 | 6582.60 | 6582.60 | under 0.001 mm |
Crossed belts, and what this calculation leaves out
Crossing the belt reverses the driven shaft and wraps both pulleys equally, through π + 2α with sin α = (D + d) ÷ 2C. Both wrap angles exceed 180°, so a crossed belt grips better than an open one on the same pulleys — and it needs a larger minimum centre distance, since the pulleys must clear each other by more than half the sum of their diameters rather than half the difference. The cost is that the belt rubs against itself where it crosses, which is why crossed drives are rare outside slow machinery.
Three things this page does not model. Belt thickness: the belt bends about its pitch line, so the effective diameters are slightly larger than the pulley diameters and the real ratio drifts a little from D₁ ÷ D₂. Creep: even with no gross slipping, the belt stretches on the tight side and relaxes on the slack side, losing typically 1–2% of the theoretical speed under load. And centrifugal tension: at speed, part of the belt tension goes into holding the belt on its own curved path rather than transmitting power, which is the real reason for the speed limit rather than any strength issue.
The tension ratio shown is likewise an upper bound on grip, not a design figure. It is the classic capstan result for a flat belt on a round pulley, taken at the smaller wrap angle. A V-belt wedges into its groove and behaves as though µ were much larger — the usual substitution is µ ÷ sin(β ÷ 2) for a groove half-angle β — so a real V-belt drive holds considerably more than the flat-belt number suggests.