What belt length actually means
A belt wrapped round two sheaves traces four pieces: two straight runs on the common tangents, an arc round the large sheave that is a little more than half its circumference, and an arc round the small sheave that is a little less than half of its own. Add the four and you have the belt length. Everything else in belt geometry is bookkeeping about which length you mean.
A V-belt has three lengths, and confusing them is the most expensive mistake on this page:
- Inside length is what a tape round the inside of a classical A, B, C or D belt reads, and it is the number stamped on the belt. It is the smallest of the three.
- Pitch or datum length is measured on the belt's tension member, part way up the cross-section, where the belt neither stretches nor compresses as it bends. This is the length that belongs in the geometry, because it is the length that matches the sheave pitch diameters.
- Outside or effective length is measured on the outer surface. Narrow 3V, 5V and 8V belts are designated by effective length in tenths of an inch — a 5V1000 is nominally 100.0 in.
The calculator works in pitch length throughout, and converts to the number stamped on your belt using the published section allowance. Mix pitch diameters with inside lengths and you will be out by an inch or three, which on a short drive is the whole of the motor's adjustment travel.
Belt length depends only on the pair of diameters and the centre distance, not on which sheave drives, so this page is equally correct for a reduction and for a step-up.
The exact formula, and the one written on the shop wall
Drop a line between the two shaft centres, length C. The common tangent the belt runs on is offset from that line by an angle β, and simple trigonometry gives sin β = (D − d) / (2C). Write h for that half-difference, (D − d)/2. Then:
- Each straight run is √(C² − h²), by Pythagoras on the right triangle with hypotenuse C and one side h. There are two of them.
- The large sheave carries an arc of π + 2β radians, so it contributes (D/2)(π + 2β).
- The small sheave carries π − 2β radians, contributing (d/2)(π − 2β).
Add the two sheave terms and the 2β parts collect into a single β(D − d), leaving the compact exact form: L = 2√(C² − h²) + (π/2)(D + d) + β(D − d). The middle term is the belt going half way round both sheaves; the last term is the extra the large sheave takes because it wraps more than half. When D = d, β is zero and the whole thing collapses to 2C + πD, which is the obvious answer for equal wheels.
The shop approximation — L ≈ 2C + 1.5708(D + d) + (D − d)²/(4C) — comes from expanding the exact form as a series and keeping the first two terms. It is remarkably good: for a 10 in and 4 in pair at a 20 in centre distance it lands within a thousandth of an inch. It only starts to drift when the diameter difference approaches the centre distance, which is exactly the regime where the wrap angle is already too small to run. Its real virtue is that it can be inverted algebraically, which is how the reverse formula C = (b + √(b² − 32(D − d)²))/16 with b = 4L − 2π(D + d) arises. This calculator uses the approximation only as a cross-check and solves the exact equation numerically in both directions.
The wrap angles fall out for free. The small sheave sees 180° − 2β and the large sheave sees 180° + 2β. They always sum to 360°, which is a quick way to sanity-check any belt geometry result.
Worked example: a 10 in and 4 in pair on a 20 in centre distance
You are replacing a belt on a shop machine. The large sheave is a 10 in pitch diameter, the small one is 4 in, the shafts sit 20 in apart at the middle of the motor slots, and the old belt is a B section.
- Half the diameter difference. h = (10 − 4) ÷ 2 = 3 in.
- Straight runs. √(20² − 3²) = √(400 − 9) = √391 = 19.77372 in each, so 2 × 19.77372 = 39.54744 in of straight belt.
- Offset angle. β = arcsin(3 ÷ 20) = arcsin(0.15) = 0.1505682 rad, which is 8.6269°.
- Half-wrap term. (π ÷ 2)(10 + 4) = 1.5707963 × 14 = 21.99115 in.
- Extra wrap on the large sheave. β(D − d) = 0.1505682 × 6 = 0.90341 in.
- Total. 39.54744 + 21.99115 + 0.90341 = 62.442 in pitch length.
- Convert to the stamped number. A B-section belt's pitch length runs 1.8 in above its inside length, so the inside length is 62.442 − 1.8 = 60.642 in. The nearest stock size is a B60, whose 61.8 in pitch length is 0.642 in shorter than the geometry asks for.
- Adjust the centre distance. Shortening the belt by 0.642 in pulls the shafts together by roughly half that, since the two straight runs both shorten — the exact solution is a centre distance of about 19.68 in. Well inside the travel of an ordinary motor base.
- Check the wrap. 180° − 2 × 8.6269° = 162.75° on the small sheave, comfortably above the 120° floor.
Compare the shop approximation on the same numbers: 2 × 20 + 1.5708 × 14 + 6² ÷ (4 × 20) = 40 + 21.9912 + 0.45 = 62.4412 in. It differs from the exact 62.4420 in by 0.0008 in — about a fiftieth of the width of a pencil line.
How to read the result
The centre distance is the number you act on. Belt lengths come in steps, so the geometry almost never lands on a stock size. Take the nearest catalogue belt and move the motor to the centre distance shown; that is the whole design decision. As a rough guide, changing the centre distance by one inch changes the belt length by about two inches, because both straight runs change together.
Leave room in both directions. The centre distance you calculate is where the drive runs, not where it is assembled. You need to be able to shorten the centre distance enough to drop the belt into the grooves by hand — never lever a V-belt over a sheave flange, which breaks tension cords invisibly — and then to lengthen it enough to take up the initial seating stretch and the slow growth over the belt's life. Belt makers publish an installation allowance and a take-up allowance for each section and length; check yours and confirm the motor slots cover both.
Wrap angle on the small sheave is the health check. At 180° a V-belt carries its full catalogue rating. As wrap falls the arc-of-contact correction factor cuts that rating, and design practice keeps the wrap at 120° or more. If your result is below that, the cures are a longer centre distance, a smaller diameter ratio, or a backside idler on the slack side.
Ratio, centre distance and wrap are one system. You cannot fix a short wrap by tightening the belt, only by changing geometry. If the ratio is fixed by the speed you need — work that out first with the pulley speed calculator — then the centre distance is your only lever, and if there is no room for it, the answer is a two-stage drive.
The distance limits are rules of practice, not physics. Design manuals advise keeping the centre distance at or above the large sheave diameter and no more than about three times the sum of the two diameters. Short spans bend the belt too many times a minute; long spans let the slack side whip and beat itself to pieces. Both bounds are advisory, and both are worth respecting.
V-belt sections, top widths and length conventions
| Section | Nominal top width | Designated by | Add to reach pitch length | Usual stock range |
|---|---|---|---|---|
| A | 1/2 in | Inside length | +1.3 in | 26 to 128 in |
| B | 21/32 in | Inside length | +1.8 in | 35 to 300 in |
| C | 7/8 in | Inside length | +2.9 in | 51 to 420 in |
| D | 1-1/4 in | Inside length | +3.3 in | 120 to 660 in |
| 3V | 3/8 in | Effective length | — | 25 to 140 in |
| 5V | 5/8 in | Effective length | — | 50 to 355 in |
| 8V | 1 in | Effective length | — | 100 to 500 in |
Section allowances and stock length ranges follow the classical and narrow V-belt tables in Machinery's Handbook. Individual suppliers stock subsets of these ranges, and some publish datum lengths that differ slightly from the older pitch-length convention, so confirm against the catalogue you are ordering from.
Mistakes that produce a belt that does not fit
- Mixing inside length with pitch diameter. The geometry needs pitch length and pitch diameters. Using the stamped inside length in the formula makes the calculated centre distance short by roughly half the section allowance.
- Using sheave outside diameters. A V-belt rides on the pitch line inside the groove, not on the rim. Outside diameters overstate both diameters and therefore the belt length.
- Measuring the centre distance at the end of the slots. Measure at the mid-point of the motor's adjustment travel, so you have take-up in both directions.
- Ignoring the wrap angle on a high-ratio drive. A 6:1 drive on a short centre distance can drop below 120° of contact, at which point the belts are being asked for more than they are rated to give and will burn their sidewalls.
- Levering the belt on. Rolling a V-belt over a flange snaps individual tension cords. The damage is invisible and the belt fails early. Always shorten the centre distance instead.
- Mismatching belts in a multi-groove drive. Two belts of the same nominal length can differ enough that one carries most of the load. Buy a matched set, or buy a banded belt.
- Forgetting that an idler changes everything. An idler adds its own wrap and its own tangent runs. This calculator solves the two-wheel case; a three-wheel path has to be laid out geometrically or measured with a string.
Chains, timing belts and flat belts
Roller chain uses the same wrap geometry but is constrained to whole pitches, and an even number of pitches avoids an offset link. The chain length calculator handles the rounding and reports the centre distance the resulting chain needs.
Synchronous belts are specified by pitch and tooth count. The length arithmetic is identical once you convert tooth count to pitch diameter — teeth times pitch, divided by π — but the answer must be rounded to a stock tooth count, and the centre distance follows from it. There is no equivalent of tensioning a V-belt to take up a mismatch.
Flat belts follow exactly the formula on this page, since the geometry does not care about the cross-section. What changes is the tensioning: a flat belt relies on friction over the wrap alone, with none of the wedging action a V-groove provides, so it needs both a larger wrap and higher installed tension for the same power.
If the sheave sizes are still open, decide them first from the speed you need, then fix the centre distance, then pick the belt. Reversing that order is how drives end up with the motor at the end of its slots. The gear ratio calculator and the gear centre distance calculator solve the equivalent problems for a geared drive, where the centre distance is fixed by the gears rather than chosen.
Key terms
- Pitch length
- Belt length measured on the tension member, the neutral axis of the belt in bending. It is the length that matches sheave pitch diameters, and the one the geometry uses.
- Datum length
- The modern name for what older tables call pitch length on classical belts, measured against a defined datum groove. Values differ slightly from the old pitch-length convention, so use the convention your supplier publishes.
- Centre distance
- The distance between the two shaft axes. It is the only variable you can adjust after the sheaves and belt are chosen, which makes it the design output rather than an input.
- Wrap angle (arc of contact)
- The angle of belt in contact with a sheave. 180° on both wheels is the ideal; the small sheave always has less, and a belt's power rating is derated as it falls.
- Take-up
- The extra centre distance travel reserved to tension the belt after it seats and stretches. It is separate from, and additional to, the installation allowance needed to fit the belt in the first place.