How to use the Belt Length Calculator
- 1. Choose the driveOpen belt (same direction) or cross belt (opposite direction).
- 2. Enter the geometryBoth pulley diameters and the centre distance in mm.
- 3. Read the lengthExact and approximate length, contact angle and speeds.
Belt length formulas
With radii r1, r2 and centre distance x, the textbook formulas are:
- Open belt: L = π(r1 + r2) + (r1 − r2)² / x + 2x
- Cross belt: L = π(r1 + r2) + (r1 + r2)² / x + 2x
The exact geometry uses sin α = (r1 ∓ r2) / x. Angle of contact on the smaller pulley of an open belt is 180° − 2α; for a cross belt it is 180° + 2α on both pulleys. The speed ratio is N2 / N1 = (d1 + t) / (d2 + t), reduced by any slip.
Worked example and exam tips
Pulleys of 450 mm and 200 mm on shafts 1.95 m apart with a cross belt: L = π(225 + 100) + 325² / 1950 + 2 × 1950 = 1021.0 + 54.2 + 3900 = 4975 mm. sin α = 325 / 1950, α = 9.59°, so θ = 199.2°.
- Cross belt length depends only on r1 + r2, so stepped cone pulleys keep the same belt if the sum stays equal.
- Use the smaller angle of contact when finding belt tension ratio (T1/T2 = e^(μθ)).
- Keep θ in radians in the tension formula.
Frequently asked questions
Which formula should I use in exams?
Most Indian textbooks give the approximate formulas shown first. The exact value is shown too, and the two are usually within a few millimetres.
Why does a cross belt have a larger angle of contact?
The belt wraps around more of each pulley, which is why cross belts can carry more power but wear faster at the crossing.
Does belt thickness matter?
For speed ratio, use pitch diameters (d + t). Enter the thickness to include it, or leave 0.
Is this calculator free?
Yes. You enter your name and WhatsApp number once to see your result, and all other Uncodemy tools then open without the form. Everything is calculated in your browser.