Science & Engineering
Belt wrap angle.
Wrap angle of an open belt on the small pulley, φ = π − 2·arcsin((D − d) ÷ 2C), and on the large pulley, from the two diameters and the centre distance. Optionally find the centre distance for a target angle. Runs in your browser.
Pulleys
Result
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Worked example
A 300 mm pulley drives a 100 mm pulley with the shafts 500 mm apart: sin α = (D − d) ÷ 2C = 200 ÷ 1000 = 0.2, so α = 11.54°. The small-pulley wrap is 180° − 2 × 11.54° = 156.93° and the large-pulley wrap is 180° + 2 × 11.54° = 203.07°. The belt touches 136.9 mm of the small pulley's rim.
How it works
For an open belt the two straight spans are not parallel when the pulleys differ in size. Each leaves the pulleys at an angle α to the line of centres, where sin α = (D − d) ÷ 2C. The belt therefore wraps the small pulley through π − 2α and the large pulley through π + 2α; the two always add to 360°. The small-pulley figure is the one that limits grip, which is why it is quoted first.
To find the centre distance for a wanted wrap φ on the small pulley, the same relation is turned round: C = (D − d) ÷ (2·sin((π − φ) ÷ 2)). If the answer is not greater than half the sum of the diameters, the pulleys would touch and the angle cannot be reached. Belt contact length is simply wrap angle (in radians) times pulley radius.
Not for safety-critical design
Planning and learning aid only. It is not for structural, lifting, pressure-system or other safety-critical design, and it is not professional engineering advice. Have a qualified engineer check anything that carries people or could fail dangerously.
Sources and basis: tec-science: Belt drive, wrap angle of the small pulley; Machinists' and Draftsmen's Handbook (Lobben), belt arc of contact, Project Runeberg. Checked 2026-10-16. Planning aid only; see the note above.
FAQ
What is the belt wrap angle formula?
For an open belt, the wrap angle on the small pulley is φ = π − 2·arcsin((D − d) ÷ 2C), where D and d are the large and small pulley diameters and C is the centre distance. The large pulley gets π + 2·arcsin((D − d) ÷ 2C).
Why does the small pulley matter most?
It has the smaller arc of contact, so it is where the belt is most likely to slip. A larger centre distance or a smaller size difference raises the wrap.
How do I increase the wrap angle?
Move the pulleys further apart, use pulleys closer in size, or add an idler that pushes the belt onto the small pulley.
What is the minimum wrap angle?
It depends on the belt type and the maker. Check the belt manufacturer's data; this calculator does not state a limit.
Does this work for crossed belts?
No, it is for open belts. Use the Pulley Belt Length Calculator for crossed-belt length.