SizingKitFree belt length calculator
A belt is two spans and two arcs
Give it two pitch diameters and a center distance and it returns the belt pitch length; give it the belt you already own and it returns the center distance that takes it. Both come from the real geometry — the two straight runs between tangent points plus the arc on each pulley — with the familiar 1.57 approximation printed beside the answer so you can see the gap instead of inheriting it. Wrap angle, speed ratio, belt speed and the stamped length to order come with it. Free, no signup, all of it computed in the page.
- 100% free
- No signup
- Exact arc geometry
- Both directions
- in and mm
Pitch diameter, not the rim you can measure — a sheave is stamped with it.
This is the one that sets both the wrap and the belt life.
Shaft center to shaft center, at the position the mounts are set to now.
Only the belt speed and the driven speed use this. The length does not.
Default 6,500 ft/min — The rim speed belt-drive design manuals give for ordinary cast-iron sheaves. It is not a standard's limit and there is no single one — it is the point above which the manuals send you to a ductile iron or steel sheave.
The allowance turns the stamped inside length into the pitch length this page works in.
BELT PITCH LENGTH
62.44 in
1,586.0 mm on the pitch line
WRAP ON THE SMALL PULLEY
162.7°
2.840 rad · 197.3° on the large one
- Speed ratio
- 2.5000:1
- Driven shaft
- 700 rev/min
- Belt speed
- 1,833 ft/min / 9.31 m/s
- Straight span each side
- 19.774 in / 502.3 mm
- The series formula would have said
- 62.441 in — 1 thou short
- Belt to order in section B
- stamped inside length nearest 60.6 in
Exact geometry, and the shop formula beside it. The length above is two straight spans plus the two arcs the belt actually lies on, not the series approximation, and the row above prints what that approximation would have given so the difference is visible instead of inherited. The gap grows with the diameter difference and shrinks with the center distance, which is why it is invisible on a long drive and worth a look on a short one with a big ratio. Nothing here corrects for belt thickness: these are pitch-line figures, and a pitch diameter is not the rim you can put a tape on.
Classical sections, and the belt to ask for at 62.44 in pitch length
Cross sections in inches and millimeters, then what a 62.44 in pitch length is stamped as in each one. The stock list steps in one and two inch increments, so the center distance has to be adjustable either way.
| Section | Top width | Height | Inside → pitch | Nearest stamped length |
|---|---|---|---|---|
| A | 0.500 in (12.7 mm) | 0.313 in (7.9 mm) | add 1.3 in | 61.1 in inside |
| B | 0.656 in (16.7 mm) | 0.406 in (10.3 mm) | add 1.8 in | 60.6 in inside |
| C | 0.875 in (22.2 mm) | 0.531 in (13.5 mm) | add 2.9 in | 59.5 in inside |
| D | 1.250 in (31.8 mm) | 0.750 in (19.1 mm) | add 3.3 in | 59.1 in inside |
| E | 1.500 in (38.1 mm) | 1.000 in (25.4 mm) | add 4.5 in | 57.9 in inside |
Classical V-belt cross sections A through E, RMA/MPTA IP-20 and ISO 4184. The allowance is the figure a belt maker's catalog adds to the stamped inside length to reach the pitch (datum) length that the drive geometry is calculated on. A classical belt is marked with its nominal inside length — a B75 is 75 in inside and 76.8 in on the pitch line — so the number on the belt is not the number this page works in. Narrow-section 3V, 5V and 8V belts are marked with their effective length in tenths of an inch instead, and take no allowance at all.
How to get the belt length, or the center distance, right the first time
Three inputs, two of which are usually written on the sheave and one of which is where the motor happens to be sitting.
Say which end of the problem you have
Designing a drive means going from a center distance to a belt; fixing one means going from the belt in your hand to where the motor base has to end up. Both directions are the same geometry, and the second is solved by narrowing in on the center distance that reproduces your belt length exactly rather than by the quadratic that inverts the shop formula.
Enter both pitch diameters, larger first
Pitch diameter, not the rim you can put a tape across — a V-belt rides above the bottom of its groove, so the effective circle is inside the outside diameter by a few tenths of an inch and the sheave is stamped with the figure you want. Inches, fractions and millimeters are all read, and the panel says so if the two are entered the wrong way round, because the wrap angles depend on which one is the small one.
Read the wrap and the belt speed before you accept the length
A length that fits is not automatically a drive that works. Wrap on the small pulley under about 120° puts you into the arc-of-contact correction in the maker's rating table, and rim speed past the figure in the limit field is a sheave question rather than a belt question. Both are printed beside the length, and the note underneath names whichever one is closest to deciding your drive.
Technical specifications
| Geometry solved | Two tangent spans plus both arcs — L = 2√(C² − ((D − d)/2)²) + R(π + 2α) + r(π − 2α), not the series approximation |
|---|---|
| Approximation shown | The 1.57 form printed alongside with the gap in thousandths: 0.0008 in on a 10 and 4 pair at 20 in centers, 0.047 in on a 15 and 4 pair at 12 in |
| Reverse solution | Center distance found by 80 bisection steps on the exact length, which converges below a millionth of an inch on any drive this page accepts |
| Refusals | A center distance below (D + d) ÷ 2 has the rims overlapping and a belt shorter than the wrapped-together path cannot be fitted — both are refused with the limiting figure, not approximated |
| Wrap angle | Printed for both pulleys in degrees and radians, with 120° on the small one flagged as the point the maker's arc-of-contact factor starts to matter |
| Belt sections | A through E with top width, height and the inside-to-pitch allowance of 1.3, 1.8, 2.9, 3.3 and 4.5 in — RMA/MPTA IP-20 and ISO 4184 |
| Input ranges | Pitch diameters 0.5 to 120 in, center distance 0.5 to 600 in, belt length 5 to 1,200 in, shaft speed 1 to 30,000 rev/min |
| Nothing leaves the tab | The bisection, the arcs and the section table all run locally, so a drive you are quoting is not typed into anybody else's server |
Frequently asked questions
Is the number stamped on my belt the length this page wants?
Not for a classical V-belt. An A, B, C, D or E belt carries its nominal inside length — a B75 is 75 in around the inside — while the drive geometry is calculated on the pitch line, which sits up inside the section where the tensile cords run. The allowance between the two is 1.3 in for an A, 1.8 in for a B, 2.9 in for a C, 3.3 in for a D and 4.5 in for an E, so that B75 is 76.8 in of pitch length. Narrow-section belts are different again: a 5V1250 is marked with 125.0 in of effective length directly and takes no allowance at all, which is one of the quieter reasons a drive gets rebuilt with the wrong belt.
How wrong is the 1.57 formula everyone uses?
Wrong by thousandths on an ordinary drive and by enough to notice on a short one with a big diameter difference. It is a two-term series expansion of the exact geometry, and its error grows with the diameter difference and shrinks with center distance: a 10 in and 4 in pair at 20 in centers comes out 0.0008 in short, which is nothing, while a 15 in and 4 in pair at 12 in centers is 0.047 in short — half a millimeter, on a drive where the take-up range might be an inch. The panel prints both figures so you can see which situation you are in rather than assuming.
How much center-distance adjustment does the drive need?
Enough to get the belt on slack and then enough again to take up what it loses in service, and the two are separate allowances in the same slot. A new V-belt seats into its grooves and stretches over the first hours of running, which is why manufacturers ask for a re-tension after the run-in and why a fixed-center drive needs a spring-loaded or manually set idler instead of a movable motor base. Stock belt lengths step in one and two inch increments through the useful range, so a drive designed at exactly 62.44 in gets built with the belt above or below it and the center distance moved to suit — which is the practical reason this calculator solves in both directions.
What happens if the wrap on the small pulley drops below 120°?
The belt runs out of contact to grip with, and the maker's rating table takes the power capacity down accordingly. Every V-belt rating is published for 180° of wrap, with an arc-of-contact factor applied below it; the factor falls slowly at first and then steeply, and around 120° most design manuals stop recommending the drive at all. This page does not reproduce that factor because it belongs to the maker and varies by section and construction, so it tells you the wrap angle and the geometry that produced it, and the catalog for the belt you are buying supplies the correction. Moving the shafts apart is the cheapest fix, since wrap rises with center distance for the same pair of pulleys.
Does this work for a crossed belt or a three-pulley layout?
No — it solves the open drive only, where both pulleys turn the same way and the belt does not cross itself. A crossed belt is a different formula: the arcs both become π plus the offset instead of one plus and one minus, so the length is 2C + 1.57(D + d) + (D + d)² ÷ 4C, with a plus sign where the open drive has a minus. A serpentine or three-pulley path is not a formula at all — it is a sum of tangent segments and arcs around each pulley in the order the belt meets them, and the tangent points depend on which side of each pulley the belt runs, so it has to be laid out rather than looked up.
My sheave is not stamped. Can I measure the pitch diameter?
You can get close, and close is usually good enough for a length but not for a ratio you are relying on. Measure the outside diameter and subtract roughly twice the depth from the top of the groove down to the pitch line, which is a fraction of an inch on an A or B section and grows with the section. The error is worse than it looks on a small sheave, because a tenth of an inch out of a 3 in pitch diameter is 3% of the ratio while the same tenth out of a 12 in one is under 1%. If the drive matters, read the part number off the sheave and look the pitch diameter up.
Why does belt speed matter if the length is already fixed?
Because the power a belt can carry is its tension times its speed, and both ends of that have limits. Too slow and the drive needs impractical tension for the power, which is why a large slow sheave on a small motor is a bad way to build a reduction; too fast and centrifugal force starts unloading the belt from its grooves while the sheave itself approaches its rim speed rating. The 6,500 ft/min in the limit field is the customary figure for cast-iron sheaves and is editable because it belongs to the sheave rather than the belt — ductile iron and steel are rated higher, and the sheave's own catalog is the authority.
About two-pulley geometry, and the formula that quietly rounds
The belt path around an open drive is exactly four pieces: a straight run down each side, tangent to both pulleys, and the arc each pulley holds. Everything else follows from one angle — the tilt of those straight runs against the line between the shaft centers, which is the arcsine of half the diameter difference over the center distance. The large pulley then holds π plus twice that angle and the small one holds π minus it, which is the whole reason a reduction always runs short of contact at the end where the belt is most loaded. Add the four pieces and you have the length, with no approximation anywhere in the arithmetic and no term dropped to make it fit on a slide rule.
The formula printed on every other calculator, L = 2C + 1.57(D + d) + (D − d)² ÷ 4C, is that same geometry with the arcsine expanded as a series and everything past the first correction thrown away. It was a good trade in 1950 and it is still accurate to within a thousandth of an inch on a normal drive. What it does not tell you is when it stops being accurate, and the answer is exactly when you least want it to: a big diameter difference over a short center distance, which is the compact reduction somebody is checking by hand because it looks tight. Printing both numbers costs a line of a table and settles the question. The reverse direction is worth more still — the quadratic that inverts the approximation inherits its error and adds a square root to it, while a bisection on the exact length converges to well under a thousandth in eighty steps and takes no longer than a keystroke.
What this page deliberately does not do is tell you the drive will carry the load. A V-belt's power rating is a table per section, per small-sheave diameter, per rev/min, multiplied by a correction for arc of contact, another for belt length and a service factor for what the machine does to it — that table belongs to the belt maker and no calculator should be inventing one. Geometry is what can be computed honestly, and it is what decides whether the drive can be built at all. Where the ratio has to be exact rather than merely close, teeth do what a friction drive cannot, and the gear train page works in whole numbers for that reason. The torque the two shafts are carrying is a separate conversion from the speed ratio, an idler held against the slack side is sized by its spring rate rather than by its diameter, and the jackshaft carrying an overhung sheave is a cantilever with a load on the end long before it is a drive component.
Where the drive you are laying out is worked out
Every number on this page is worked out by JavaScript running in the tab you are reading it in. Nothing you type — loads, lengths, nameplate ratings, the rates your utility charges you — is uploaded, logged or kept, which is also why the calculators carry on working in a mechanical room with no signal.
The eighty bisection steps behind the reverse solution run in your browser between one keystroke and the next, so the diameters and center distances of a machine you are rebuilding are never sent anywhere and there is nothing kept to come back to. Print the section table if you want a copy — that is the only record this page can make.