Calculate the wrap angle θ at the small pulley of an open belt drive from both pulley diameters and the centre distance. It is always below 180°, limits the transmittable belt force and is therefore the governing value at the small pulley.
Select a target and calculate.
θ = π − 2·arcsin((D − d)/(2·C)) approaches 180° (dashed) as the centre distance grows. Short centre distances and large diameter differences reduce the wrap on the small pulley and hence the transmissible force. The curve starts where the pulleys touch, C = (D + d)/2.
Calculate the wrap angle θ at the small pulley of an open belt drive from both pulley diameters and the centre distance. It is always below 180°, limits the transmittable belt force and is therefore the governing value at the small pulley.
250/100 mm pulleys at 570 mm center distance give about 164.9° wrap at the small pulley.
Open belt drive; the wrap angle at the large pulley equals 360° minus this angle.
This calculator finds the missing quantity among wrap angle θ (at the small pulley), pulley diameters D and d, and center distance C from the other three. It shows why the wrap angle at the small pulley is always the critical figure for a belt drive's force-transmission capacity.
In an open belt drive, the belt contacts the smaller pulley over an angle θ = π − 2·arcsin((D−d)/(2C)), always below 180°. The larger pulley is correspondingly wrapped over more than 180°; the two angles sum to 360°.
Manufacturer power ratings for V-belts are, by default, based on a wrap angle of exactly 180°. If the actual angle at the small pulley deviates from that — say, due to a large diameter difference or a short center distance — the rated power must be adjusted with a wrap-angle correction factor that drops noticeably as the angle falls.
θ = π − 2·arcsin((D−d)/(2C))
θ = π − 2·arcsin((D−d)/(2C))C = (D−d) / (2·sin((π−θ)/2))D = d + 2C·sin((π−θ)/2)| Symbol / input | Meaning |
|---|---|
| Wrap angle θ (small pulley) | Angle over which the belt contacts the smaller pulley. |
| Large pulley diameter D | Pitch diameter of the larger pulley. |
| Small pulley diameter d | Pitch diameter of the smaller pulley. |
| Center distance C | Distance between pulley centers. |
Enter three of the four quantities to find the fourth. The calculated angle θ always refers to the smaller pulley d; the wrap angle at the larger pulley D follows as 360° − θ.
Select the target quantity and enter the three known values with units. Compare the calculated angle against your belt type's manufacturer wrap-angle correction factor afterward if it deviates noticeably from 180°.
Given D = 250 mm, d = 100 mm and C = 570 mm, substitution gives θ = π − 2·arcsin(150/1,140) ≈ 164.9° at the small pulley.
A wrap angle of 164.9° sits close to the 180° used in rating tables, indicating good force-transmission capacity. At the large pulley, the angle correspondingly works out to 360° − 164.9° = 195.1°.
Angle is given in degrees (°) or radians; diameters and center distance in the same length unit, usually millimetres.
The force a belt can transmit depends exponentially on wrap angle per the capstan equation: F1/F2 = e^(μθ). Since the small pulley always has the smaller of the two angles, it limits the maximum transmittable force before the belt slips. Manufacturer tables for V-belt power capacity therefore assume a reference angle of 180°; at smaller angles, allowable power is reduced via a correction factor. Increasing center distance or adding an idler pulley can raise the wrap angle at the small pulley without changing the ratio itself.
Wrap angle is used to design belt drives with a large diameter difference, apply wrap-angle correction factors from manufacturer tables, and troubleshoot slipping belts. It is also the exponent in the capstan equation: the maximum belt tension ratio F₁/F₂ = e^(μθ) and with it the transmittable belt power depend directly on θ at the small pulley.
The formula applies to an open two-pulley belt drive without an idler pulley. An additional idler can deliberately increase the wrap angle at the small pulley and is not captured by this simple two-pulley formula.
Common mistake: Do not confuse wrap angle with center distance or diameter ratio — a short center distance with a large diameter difference noticeably lowers the angle at the small pulley even though the ratio itself stays unchanged. Also, for a very short center distance, check that (D−d)/(2C) stays below 1, since otherwise no valid angle exists geometrically.
Because the line connecting the pulley centers runs at an angle to the belt tangent when diameters differ; this angle of inclination reduces wrap at the small pulley and increases it at the large pulley by the same amount.
Because manufacturer belt power ratings are, by default, given for a wrap angle of 180°. At a different angle, allowable power is adjusted via a correction factor.
By increasing center distance or adding an idler pulley that routes the belt further around the small pulley.