A belt drive is one of the few machine elements where getting the length wrong by two percent is the difference between a drive that runs for a decade and one that will not go together at all. Belts come in fixed catalogue lengths, they stretch very little on purpose, and the tensioner usually has only a few centimetres of travel. The geometry has to be settled before anything is ordered.
This belt length calculator from Arb Digital uses the exact trigonometric expression rather than the textbook approximation, then shows the approximation beside it so you can see what the shortcut costs on your geometry. It also reports the three numbers that decide whether a drive works — wrap angle on the small pulley, free span, and belt speed — because a drive can have a correct belt length and still slip, whine or shred itself inside a month.
What This Belt Length Calculator Does
You give it two pulley diameters and a centre distance. It returns the belt length that path requires, in your chosen unit and in the other one, so you can cross a catalogue quoted the opposite way.
It handles both arrangements. An open drive is the ordinary one: the belt runs around the outside of both pulleys and both shafts turn the same way. A crossed drive twists the belt in a figure of eight so the driven shaft turns backwards, and its length formula is genuinely different rather than a small correction.
The four supporting numbers cover the things that break drives. Wrap angle on the small pulley decides how much torque friction can carry before the belt slips. Free span is the unsupported length that flutters and sets the tension you would check by plucking it. Speed ratio confirms the pulley pair gives the output speed you wanted. Belt speed decides which belt construction is even eligible.
Finally, the note underneath solves the problem backwards. Enter a length you can actually buy and the tool returns the centre distance that belt would sit at — the number you need when positioning a motor base rather than designing from scratch.
How to Use It
- Choose your unit first. Mixing a metric centre distance with imperial diameters is the commonest way to get a plausible-looking wrong answer.
- Enter pitch or datum diameters, not outside diameters. Catalogue lengths are quoted on that line, not on the outer surface.
- Enter the centre distance you have or the one you want. If you are still laying the machine out, try a value between one and three times the large diameter.
- Check the wrap angle before anything else. Below about 120 degrees on the small pulley, a friction belt loses meaningful capacity.
- Put a stock length in the last box. The note then tells you where the motor has to sit for that belt.
The Formula and a Worked Example
Take the large radius as R, the small radius as r and the centre distance as C. For an open drive the belt leaves each pulley along a common tangent, and that tangent is tilted by an angle α where sin α = (R − r) / C. The two straight sections are each √(C² − (R − r)²) long. The belt wraps π + 2α radians around the large pulley and π − 2α radians around the small one. Add the four pieces:
L = 2√(C² − (R − r)²) + R(π + 2α) + r(π − 2α)
The familiar textbook version replaces all of that with L ≈ 2C + (π/2)(D + d) + (D − d)²/(4C), which comes from expanding the exact expression as a series and keeping two terms. It is remarkably good. The tool computes both and reports the gap.
Work the default through. A 200 mm pulley drives a 100 mm pulley at 500 mm centres, so R = 100, r = 50 and R − r = 50. Then sin α = 50/500 = 0.1, giving α = 0.100167 rad. Each straight span is √(250,000 − 2,500) = 497.494 mm, so the two together are 994.987 mm. The large pulley carries 100 × 3.341927 = 334.193 mm of belt and the small one carries 50 × 2.941259 = 147.063 mm. The total is 1,476.24 mm. The approximation gives 1,000 + 471.239 + 5 = 1,476.24 mm — the two agree to four thousandths of a millimetre on this geometry.
The approximation degrades when the pulleys are very different in size and the centres are short, because the neglected terms scale with ((D − d)/C) raised to increasing powers — a 300 mm and a 40 mm pulley at 200 mm centres is where you notice it. That is precisely the geometry the exact formula exists for.
For a crossed drive the two tangents cross between the pulleys, so the offset angle uses the sum rather than the difference: sin α = (R + r) / C, each span is √(C² − (R + r)²), and both pulleys wrap the same π + 2α. A crossed belt is always longer than the open belt on the same pulleys, and it has the useful property that both pulleys get more than half a turn of contact regardless of size mismatch. The National Programme on Technology Enhanced Learning module on belt drives from IIT Kharagpur derives both cases in full; Module 13, Lesson 1, Introduction to Belt drives is the relevant chapter.
Why Wrap Angle Decides More Than Length
A friction belt transmits torque because the tight-side tension exceeds the slack-side tension, and the maximum ratio between them follows the capstan relationship: T₁/T₂ ≤ e^(μθ), where θ is the wrap angle in radians. Everything happens on the small pulley, because it always has the smaller wrap.
At 180 degrees of wrap and a friction coefficient of 0.3, the tension ratio can reach about 2.6. Drop the wrap to 120 degrees and it falls to roughly 1.9 — a third of the drive's torque capacity gone, silently. The belt does not fail; it slips a little more on every start, gets hot, glazes, then slips a lot.
V-belts cheat this with geometry. The wedge shape multiplies the effective friction coefficient by roughly 1/sin(β/2) where β is the groove angle, so a 38-degree groove gives about a threefold gain. That is the entire point of the V section. Synchronous belts avoid the question by meshing teeth, but they replace it with a minimum-teeth-in-mesh rule, typically six, which is a wrap-angle constraint wearing different clothes.
If the tool shows the small pulley below about 120 degrees you have three options: increase the centre distance, reduce the size difference by splitting the reduction into two stages the way our gear ratio calculator handles multi-stage trains, or add an idler pushing into the slack side at the cost of an extra bearing and a tighter bend radius.
Datum, Pitch and Outside Length Are Three Different Numbers
This is where most ordering errors originate, and it has nothing to do with the trigonometry. A belt has several length definitions and catalogues do not always say which one they are quoting.
Outside length is measured around the outer surface. Inside length is measured around the inner surface, and it is what many older imperial V-belt part numbers encode. Pitch or datum length is measured at the belt's neutral axis, the layer that neither stretches nor compresses as the belt bends, and only that length is geometrically meaningful — the neutral axis is the only part of the belt that actually travels the path this calculator computes. For a classical A-section V-belt, inside and datum length differ by around 33 mm, which on a 1,200 mm belt is nearly three percent and far beyond any tensioner's range. The datum system for classical and narrow V-belts is standardised — ISO 4184, Belt drives: classical and narrow V-belts, lengths in datum system is the governing document, and it lists the preferred datum lengths and their tolerances for the Y, Z, A, B, C, D, E and SP sections.
The matching rule for pulleys is to enter the diameter measured at the same line. For a V-sheave that is the datum diameter, below the top of the groove. For a timing pulley it is the pitch diameter — tooth count times tooth pitch divided by π — which is slightly larger than the outside diameter, since a synchronous belt's cords sit above the tooth tips. That means a timing pulley's pitch diameter can never be measured with callipers.
Solving Backwards for Centre Distance
Design rarely runs in the direction the formula does: you have a stock belt length and need to know where to bolt the motor. Rearranging the approximate formula gives a closed solution. Let b = 4L − 2π(D + d). Then:
C = [ b + √(b² − 32(D − d)²) ] / 16
Check it against the worked example. With L = 1,476.24 and the same pulleys, b = 4,020, the discriminant is 15,840,400, and its square root is 3,980. So C = 8,000/16 = 500 mm, which is where we started.
If the discriminant goes negative, no centre distance exists for that belt on those pulleys and you need a longer belt or smaller ones. And the computed distance is the nominal position, not the installation position: you must be able to move the centres closer to slip the belt on and further apart to tension it. Drive manuals normally ask for roughly one to two percent of belt length inwards and about three percent outwards, so a 1,500 mm belt wants around 20 mm of slack travel and 45 mm of take-up.
Belt Speed, and Why It Caps Your Options
Belt speed is the pulley circumference multiplied by rotational speed, and this tool computes it from the small pulley because that is normally the motor. The default case — a 100 mm pulley at 1,750 rpm — gives π × 0.1 × 1,750/60 ≈ 9.2 m/s, comfortably in the middle of the useful band.
Classical V-belts are happiest between roughly 5 and 25 m/s. Below about 5 m/s the belt carries a large force to move modest power, so it needs a wide section and the drive gets bulky; our horsepower calculator and torque calculator make that trade visible. Above roughly 30 m/s centrifugal force starts lifting the belt out of the groove, reducing the wedging action the V section relies on. You set that band with pulley size, not with the belt.
Where This Sits Next to Our Other Tools
One boundary is worth stating plainly, because the names collide. Our belt size calculator is about clothing: it turns a waist or trouser size into the belt size to order. This page is about machinery, and the two share nothing but a word. Within the mechanical set, the gear ratio calculator covers multi-stage speed and torque chains, the engine rpm calculator handles drivetrain speed chains, and the angular velocity calculator converts rpm into the radians per second that dynamics work needs.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering outside diameters instead of pitch or datum diameters — the error is small on each pulley but it always lands the same way, so it never cancels.
- Ordering to inside length when the catalogue quotes datum length — on a classical V-belt those differ by around 33 mm.
- Designing to the exact computed centre distance with no take-up — you cannot fit a belt onto a drive already at nominal centres, nor tension it afterwards.
- Ignoring a wrap angle under 120 degrees — the drive runs on a bench and slips under real load, because the tension ratio falls exponentially with contact angle.
- Assuming the crossed formula is the open one with a sign flipped — the wrap angles change on both pulleys, not just the tangent length.
Related Free Tools From Arb Digital
Pair this with the gear ratio calculator when the reduction needs more than one stage, and the engine rpm calculator when you are chasing an output speed through a whole drivetrain. The torque calculator and horsepower calculator cover what the drive has to carry, and the angular velocity calculator converts rpm into radians per second. For unit changes on any of these figures, use the length converter. Everything is indexed on the free online tools hub.
Frequently Asked Questions
With R and r as the large and small pulley radii and C as the centre distance, the exact length is two times the square root of C squared minus (R minus r) squared, plus R times (pi plus twice the tangent angle), plus r times (pi minus twice the tangent angle), where the tangent angle is the arcsine of (R minus r) divided by C.
Very accurate for typical drives. On the default geometry here, 200 mm and 100 mm pulleys at 500 mm centres, the approximation differs from the exact result by about four thousandths of a millimetre. It degrades when the pulleys are very different in size and the centre distance is short, which is exactly when this tool's exact calculation earns its place.
Pitch or datum diameter, because that is where the belt's tension member travels and where catalogue lengths are measured. For a timing pulley the pitch diameter is the tooth count times the tooth pitch divided by pi, which is slightly larger than the outside diameter and cannot be measured with callipers.
Around 180 degrees is ideal and roughly 120 degrees is the usual practical floor. The tension a friction belt can hold rises exponentially with contact angle, so the loss below 120 degrees is steep. Increasing the centre distance, reducing the size mismatch or adding an idler on the slack side all raise it.
Let b equal four times the belt length minus two pi times the sum of the diameters. The centre distance is b plus the square root of b squared minus thirty-two times the squared difference of the diameters, all divided by sixteen. This tool runs that whenever you fill in the stock-length field.
Because both pulleys wrap more than half a turn instead of one wrapping more and one less, and the crossing tangents run a longer diagonal path than the parallel tangents of an open drive. The offset angle uses the sum of the radii rather than the difference, which is what makes the two formulas genuinely different.
No. The belt size calculator converts a waist or trouser measurement into a clothing belt size. This page computes the length of a mechanical drive belt from pulley geometry. They share a word and nothing else.
This tool is provided for educational and reference use. Belt lengths, tolerances, installation allowances and power ratings vary by belt section and manufacturer, so confirm any figure against the drive design data for the specific belt you intend to buy before committing to a machine layout.
