How this instrument works
Loop a belt over two pulleys and you have four pieces of geometry: two straight tangent runs, a long arc hugging whichever sheave is bigger, a short arc on its partner. Make both wheels identical and bookkeeping stops there — twice centre distance in straight run, plus exactly one circumference of wrap. Unequal diameters tilt those tangents, which trims a little off each straight and adds rather more onto total wrap. Net gain is that final term, (D₁−D₂)² ⁄ 4C, and it never goes negative: for a given pulley sum and spacing, a mismatched drive always swallows more belt than an even one.
Flat leather belting ran entire mills through the nineteenth century — one engine turning a ceiling line shaft, every lathe below tapping power off it — so this was a length millwrights figured daily. John Gates put a rubber V-belt into production at his Denver works in 1917, and wedging action inside a grooved sheave multiplied grip enough that short, tightly spaced, individually motored drives became practical. Typical magnitudes now: a bench machine wants 1 to 2 metres, a car's serpentine belt roughly 2.2 m, a quarry conveyor tens of metres. Nobody moulds arbitrary lengths, though. Stock arrives in catalogue steps of about 25 mm, so treat any figure here as a starting point — compute, round to a size somebody actually sells, then shift centres to absorb what is left over.
This expression approximates, and knowing where it came from tells you when to distrust it. Exact geometry needs an angle: sin α = (D₁−D₂) ⁄ 2C says how far tangents tilt off parallel, after which straights measure 2C cos α while wraps measure π+2α and π−2α radians. Expand those trigonometric terms as series, drop everything past second order, and out falls the line above. Truncation always runs short, by close to C·k⁴ ⁄ 12 with k = (D₁−D₂) ⁄ 2C — a fourth power, so error collapses on sane drives and bites only when pulleys differ wildly across a cramped span. Two firmer limits sit behind that one. Centre distance must exceed (D₁+D₂) ⁄ 2 or your sheaves intersect, and what comes out is bare geometry: it knows nothing of stretch, tensioner travel, or slack needed to roll a belt over a rim.
- Enter the driven sheave into Larger pulley diameter. Use its pitch or datum figure — where belt cords actually sit — rather than the outside rim.
- Enter the motor sheave into Smaller pulley diameter. Swapping both entries changes nothing arithmetically, since their difference gets squared, but labels keep your notes honest.
- Measure shaft axis to shaft axis and put that into Centre distance. Rim-to-rim clearance is a different number and will cost you a belt.
- Read Belt length. Millimetres, centimetres, metres and inches are all offered, and since every term is a length, output follows whatever unit you feed in.
- Round up to the nearest stocked size, then close Centre distance by about half the surplus — two straight runs mean each millimetre of spacing buys nearly two of belt.
Worked example — a 3:1 wood-lathe headstock
A wood lathe: 300 mm pulley on its spindle, 100 mm on the motor beneath, shaft centres 500 mm apart for a 3:1 reduction. Put 0.3 into Larger pulley diameter, 0.1 into Smaller pulley diameter, 0.5 into Centre distance, and Belt length returns 1.648319 m. Three terms build that up — 1.0 m of straight run between tangent points, 0.628319 m of arc wrapped round both sheaves, and 0.02 m of correction for tilt.
Checked against exact geometry, which gives 1.648386 m, this sheet runs 0.067 mm short: four parts in a hundred thousand, comfortably inside moulding tolerance on any rubber belt. Rounding costs far more than truncation ever will. No stockist lists 1648 mm, so buy 1650 and wind your motor mount 0.9 mm closer — that 1.68 mm of surplus divides by 1.96, because both straight runs shorten together as centres come in.
Grip deserves a glance too. Here sin α = 0.2, so α = 11.54°, and the motor pulley — smaller, hence first to slip — still keeps 156.9° of wrap. Comfortable. Under roughly 120° a V-belt starts surrendering torque capacity, and that threshold has a tidy form worth remembering: wrap holds at 120° or better whenever centre distance is at least as large as your diameter difference.
Questions
Does it matter which pulley I call the larger one?
Arithmetically, no. Their sum is symmetric and their difference gets squared, so swapping both entries returns an identical length — a handy self-check when you are unsure which sheave is which. Physically it matters a great deal. Wrap angle, and therefore how much torque passes before slipping starts, is fixed by whichever pulley is smaller. Keep to the convention so any wrap-angle or speed-ratio work alongside this reads correctly.
Should I measure outside diameter or pitch diameter?
Pitch diameter — called datum or effective diameter depending on which standard your sheave was cut to. A V-belt wedges into its groove and rides on the flanks, so its load-carrying cords sit below the rim rather than on it. Feeding outside diameters in overstates both wheels, and because arc length scales with π, an error of 5 mm per sheave inflates your answer by roughly 16 mm — enough to land one catalogue size out. Flat belting is the exception: there the outer surface is the working surface, so outside diameter is correct.
How accurate is this approximation?
Short, always, and by less than most people expect. Truncation error sits near C·k⁴ ⁄ 12 where k = (D₁−D₂) ⁄ 2C. On the worked example, k = 0.2 and the shortfall is 0.067 mm on a 1.65 metre belt. Push k out to 0.4 — a far more lopsided drive — and it grows only to about 1.1 mm. Fourth powers punish extremes and forgive everything else, so at ordinary industrial ratios this formula is finer than anyone's ability to measure centre distance with a tape.
What changes for a crossed belt?
One sign. A crossed belt figure-eights between pulleys to reverse rotation, and its length is 2C + π(D₁+D₂) ⁄ 2 + (D₁+D₂)² ⁄ 4C — difference becomes sum, which makes that correction term far bigger. Wrap improves as well, landing identically on both wheels, which is why crossed flat belts held on so willingly. They also rub against themselves where runs intersect, so V-belts and toothed belts are never crossed. This sheet handles open drives only.
How close together can two pulleys sit?
Hard floor first: centre distance must exceed (D₁+D₂) ⁄ 2 or your sheaves physically intersect, and anything at or below zero is refused outright. Grip fails well before that geometric wall. Squeezing centres in tilts a belt further, strips wrap off the smaller pulley and drops torque capacity; holding centre distance at least equal to (D₁−D₂) keeps that wrap at 120° or better. Very long spans bring separate trouble, since unsupported belt whips and flaps, so drives usually stay inside about three times the sum of both diameters.
Does this figure include tension, stretch or take-up?
No. What comes out is a static geometric length along a belt's pitch line, with pulleys exactly where you said, carrying no preload. Real installations need room in both directions: slack to roll a belt over the rim while fitting, then travel the other way to tension it and to chase the seating and elongation that arrive during a new belt's first hours of running. Motor slides and jockey pulleys exist for precisely that. Take the length from here, then confirm your mount can actually move.