Belt wrap angle at the small pulley

What does this quantity describe?

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.

Formula and variables

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)

Choose the inputs correctly

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.

How to use the calculator

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°.

Worked example

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.

Understand the result and units

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.

Why is the wrap angle at the small pulley decisive?

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.

Typical applications

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.

Assumptions, limits and common mistakes

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.