Belt Length Calculator

What this calculator measures

This is the geometric path of an open, uncrossed belt around two pulleys. The shafts are parallel, the pulley grooves are aligned and the belt has two straight external-tangent spans. It is not a serpentine-drive solver, a crossed-belt calculator or a tension calculator.

Use one consistent reference throughout. For a synchronous belt, length is specified along the pitch line and must equal a whole number of tooth pitches. For a V-belt, a catalogue may specify datum length, effective length, inside length or outside length. Those are not interchangeable names for the same circumference. Gates product data, for example, can list effective and outside circumference separately, while Optibelt drive worksheets ask for the belt type, pulley diameter, center distance and stated belt length.

Exact open-belt formula

Let:

• D = larger pulley reference diameter
• d = smaller pulley reference diameter
• C = shaft center distance
• Δ = (D − d) ÷ 2
• α = asin(Δ ÷ C) in radians

The exact tangent-and-arc length is:

L_exact = 2√(C² − Δ²) + π(D + d) ÷ 2 + (D − d)α

The first term is the two straight spans. The remaining terms are the unequal arcs of contact around the pulleys. The underlying tangent geometry requires C > |D − d| ÷ 2; a real non-overlapping drive normally also has C > (D + d) ÷ 2.

Standard approximation

A common design approximation is:

L_approx = 2C + π(D + d) ÷ 2 + (D − d)² ÷ (4C)

It is the practical expression used for preliminary two-pulley sizing. Its error grows as the pulley-diameter difference becomes large relative to the center distance, so use the exact value when comparing close stock sizes. With equal pulleys, both formulas reduce to L = 2C + πD.

Worked calculation

For D = 200 mm, d = 100 mm and C = 500 mm:

Δ = 50 mm and α = asin(0.1) = 0.100167 rad

L_exact = 2√(500² − 50²) + π(200 + 100) ÷ 2 + 100(0.100167)

L_exact = 1,476.24 mm

The approximation gives 1,476.24 mm after rounding to two decimals. That agreement is specific to this geometry; it is not permission to treat all belt-length conventions as equivalent.

Check vectors

From a calculated path to a real belt

Do not add a generic tension percentage. Installation allowance, take-up travel, measuring load, belt construction and permitted pre-tension depend on the exact product and drive. Choose a standard belt whose stated length convention matches your pulley reference, then recalculate the achievable center distance or provide manufacturer-approved adjustment. A timing belt must also have a whole tooth count: L = teeth × pitch.

Never convert between inside, outside, datum, effective or pitch length by guessing a universal offset. The difference depends on belt section and construction. Likewise, a pulley’s outside diameter may not be its datum or pitch diameter. Look up the conversion for the selected belt/pulley family.

For final selection, consult the manufacturer’s engineering resources:

• Gates Timing Belt Theory explains pitch-line length, tooth pitch and two-pulley geometry.
• Gates Belt Sizing Guide covers stock tooth counts, fixed center distances and pre-tension design.
• Optibelt Drive Calculation Form shows the product and drive details required for a manufacturer calculation.
• Gates Design Power supports product-specific belt length, tension and shaft-load design beyond this geometry estimate.