Belt length calculator (open and crossed belts)
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Result
Calculate the length of a belt around two pulleys, for an open or a crossed belt drive. The calculator gives the exact length, the common approximation and the angle of wrap on each pulley, or works backwards from a belt length to the centre distance.
Step-by-step solution
Angle of the straight belt spans
Exact belt length
Common approximation
Angles of wrap
Key formulas
- Open belt
- L ≈ 2C + π(D1 + D2)/2 + (D1 − D2)²/(4C)
- Crossed belt
- L ≈ 2C + π(D1 + D2)/2 + (D1 + D2)²/(4C)
- Speed ratio
- n1·D1 = n2·D2
How to calculate belt length
- Use the radii R (large pulley) and r (small pulley) and the centre distance C.
- Open belt: sin φ = (R − r)/C, and L = 2√(C² − (R − r)²) + π(R + r) + 2φ(R − r).
- Crossed belt: sin φ = (R + r)/C, and L = 2√(C² − (R + r)²) + (R + r)(π + 2φ).
- Approximation: L ≈ 2C + π(D₁ + D₂)/2 + (D₁ ∓ D₂)²/(4C), with − for an open belt and + for a crossed belt.
- Wrap angles: π − 2φ on the small pulley and π + 2φ on the large pulley of an open belt; π + 2φ on both pulleys of a crossed belt.
Questions and answers
Why does the wrap angle matter?
The belt can transmit more force before slipping when it wraps further around the pulley: the tension ratio is e^(μθ). The small pulley of an open drive has the smallest wrap and usually limits the power.
When is the approximation good enough?
When the centre distance is large compared with the difference in diameters. For the default example the two lengths differ by less than 0.01%.
How do I find the centre distance for a standard belt?
Choose “centre distance from the belt length” and enter the belt length. The exact formula cannot be solved for C directly, so the calculator solves it numerically and also shows the closed-form approximation.
Why does a crossed belt reverse the direction?
The belt crosses between the pulleys, so the driven pulley turns the other way. The belt also rubs on itself where it crosses, which is why crossed belts are used only at low speeds.
Practise it yourself
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