How to calculate belt length

For an open (non-crossed) two-pulley drive — a motor sheave driving a second sheave with a single belt running around both — the belt length is fixed by just three numbers: the centre distance between the shafts and the two pulley pitch diameters. The standard closed-form approximation is accurate to a fraction of a percent for any normal drive geometry, and it is the formula the belt length calculator evaluates. This guide walks through where it comes from, how to use it correctly, and the handful of mistakes that throw the answer off.

The belt-length formula

With C the centre distance, D₁ the small pulley pitch diameter and D₂ the large pulley pitch diameter (all in the same units):

L = 2C + (π/2)(D₁ + D₂) + (D₂ − D₁)² / (4C)

Every belt loop is two straight runs joined by two curved wraps, and the formula is simply that geometry written out. Reading it term by term:

• 2C — the two straight spans. The belt leaves one pulley, crosses to the other, and comes back. If the pulleys were equal in size, the straight part of the belt on each side would be exactly the centre distance, so the two spans contribute 2C. (When the pulleys differ the spans tilt slightly and grow a touch longer; that small extra is what the third term corrects for.)
• (π/2)(D₁ + D₂) — the two wrap arcs. If the pulleys were the same size, the belt would wrap exactly half of each one (180° each), and a half circumference is πD/2. Add the two halves and you get (π/2)(D₁ + D₂). This is the curved portion of the loop.
• (D₂ − D₁)² / (4C) — the unequal-pulley correction. When the sheaves differ in size, the belt no longer wraps a clean 180° on each — it wraps more on the big pulley and less on the small one, and the straight spans run at an angle rather than parallel. This last term is the (small, positive) correction that accounts for both effects. It vanishes when D₁ = D₂ and grows as the diameters diverge or the centre distance shrinks.

Use pitch diameters, not the outer rim

The single most common error is feeding in the outside diameter of the sheave instead of its pitch (effective) diameter. The line of action of a belt is not at the rim — it is at the belt's own neutral axis where its tension cords sit. For a V-belt this matters a lot: the belt wedges down into the groove and rides below the top of the sheave, so the pitch diameter is several millimetres smaller than the measured outer diameter. For a timing belt, the pitch line runs through the centre of the cords, a defined distance above the pulley root. Using the outer diameter overstates both wrap arcs and lands you with a belt that is slightly too long. Manufacturer data sheets give the pitch (or "effective" / "datum") diameter for exactly this reason — use it.

Wrap angle on the small pulley

Belt length is only half the story. For a friction drive you also need the wrap angle — the arc of belt actually gripping the pulley — because the power a belt can transmit before it slips depends on how much of it is in contact. The small pulley is always the limiting one (it has the smaller wrap), and its angle of contact is:

θ = 180° − 2·asin((D₂ − D₁) / 2C)

With equal pulleys the term in the bracket is zero, asin(0) = 0, and the wrap is a full 180°. As the large pulley gets bigger relative to the centre distance, the straight spans steepen and the small-pulley wrap drops below 180°. As a rough guide, keep the small-pulley wrap above about 120°; below that, grip falls away and you should increase the centre distance or add an idler pulley to push the belt back onto the sheave. The belt length calculator reports the wrap angle alongside the length so you can check both at once.

Timing belts are sized the other way round

The formula above treats length as the unknown, which is right for a friction belt you can buy in almost any length. A synchronous (timing) belt is different: it comes only in discrete tooth counts, and its length is simply

L = (number of teeth) × (tooth pitch)

So in practice you compute the length your geometry wants, divide by the pitch to get a tooth count, round to the nearest standard belt, and then back-solve the centre distance that the chosen belt actually allows. Because you almost never hit an exact standard length at your ideal centre distance, the small leftover is taken up by a tensioning idler or an adjustable motor base. The timing belt calculator handles the tooth-count rounding and the resulting pulley pitch diameters for synchronous drives; the pulley calculator covers the speed-ratio and driven-RPM side of the same drive.

Worked example

Take a small pulley of D₁ = 100 mm, a large pulley of D₂ = 250 mm, and a centre distance of C = 400 mm. Work through the three terms:

• Straight spans: 2C = 2 × 400 = 800 mm
• Wrap arcs: (π/2)(D₁ + D₂) = (π/2)(100 + 250) = (π/2)(350) ≈ 549.8 mm
• Unequal-pulley correction: (D₂ − D₁)² / (4C) = 150² / 1600 = 22500 / 1600 ≈ 14.1 mm

L = 800 + 549.8 + 14.1 ≈ 1363.9 mm

So the belt is about 1364 mm long. Notice how the correction term contributes only ~14 mm of the total — about 1% — which is why some quick estimates drop it, and also why dropping it is a real (if small) error rather than a rounding nicety. Checking the small-pulley wrap for the same drive:

θ = 180° − 2·asin(150 / 800) = 180° − 2·asin(0.1875) ≈ 158.4°

That is a healthy wrap, comfortably above the 120° guideline, so this drive grips well without needing an idler.

Common mistakes

Three errors account for almost every wrong belt-length answer:

• Using outer diameter instead of pitch diameter. Especially with V-belts, the belt rides below the groove top, so the effective diameter is smaller than what your callipers read at the rim. Always work from the pitch / datum diameter.
• Ignoring the unequal-pulley term. Approximating the length as just 2C + (π/2)(D₁ + D₂) is fine for a rough idea, but it underestimates the length whenever the pulleys differ — and the gap widens as the size difference grows or the centre distance shrinks. Keep the (D₂ − D₁)² / 4C term for a proper figure.
• Confusing crossed and open belts. This formula is for an open drive, where both pulleys turn the same way. A crossed belt (the strands cross between the shafts, reversing the driven direction) has a different length expression — the diameters add in the correction rather than subtract — and it is rare in modern practice because the strands rub where they cross. Make sure you know which arrangement you have before reaching for a formula.

Get the three inputs right — pitch diameters and the true centre distance — and the formula does the rest. To skip the arithmetic, including the wrap-angle check, use the belt length calculator.

Frequently asked questions

Do I use the outside diameter or the pitch diameter of the pulleys?

Always use the pitch (effective) diameter, not the outer rim. A V-belt rides down inside the groove, so its line of action sits below the top of the sheave; a timing belt pitches at the centre of its tension cords. Using the outside diameter overstates the wrap arcs and gives a belt length that is slightly too long.

What is the wrap angle and why does it matter?

The wrap angle is the arc of belt actually in contact with a pulley. On the small pulley it is θ = 180° − 2·asin((D₂ − D₁) / 2C). It matters because grip — and therefore the power a friction belt can transmit before slipping — scales with the wrap angle. If the small-pulley wrap drops much below about 120°, you usually add an idler or increase the centre distance.

Why is timing-belt length handled differently?

A synchronous (toothed) belt only comes in discrete tooth counts, so you do not get to choose an exact length. You pick the nearest standard belt (its length is teeth × pitch), then back-solve the centre distance that the belt allows, and take up any small mismatch with a tensioning idler or an adjustable motor base.

Last reviewed: 2026-05-29.