How to use the pulley belt length calculator
- Choose your units, then enter the diameter of the large pulley and of the small pulley, measured at the groove where the belt rides.
- Enter the center distance, from the center of one shaft to the center of the other.
- Read the length of an open belt, exact and by the common approximation, and the length of a crossed belt if the shafts are far enough apart.
- Enter the speed of the small pulley in the advanced options to get the speed of the large pulley and of the belt.
Formula
open belt: theta = asin((D - d) / (2 C)), length = 2 C cos(theta) + pi (D + d) / 2 + theta (D - d)
approximation: length = 2 C + pi (D + d) / 2 + (D - d)^2 / (4 C)
crossed belt: phi = asin((D + d) / (2 C)), length = 2 C cos(phi) + (D + d) (pi / 2 + phi)
wrap angle on the small pulley = 180 degrees - 2 theta
D is the large diameter, d the small diameter and C the center distance, all in one unit. The speed of the large pulley is the small pulley speed times d / D.
An 8 inch and a 3 inch pulley, 20 inches apart
For D = 8, d = 3 and C = 20 inches, the angle theta is asin(5 / 40) = 7.18 degrees. The exact open belt length is 40 x cos(7.18) + pi x 11 / 2 + 0.1253 x 5 = 57.592 inches (the angle in radians). The common approximation gives 57.591 inches, a difference of 0.0004 inches. A crossed belt on the same pulleys is 58.801 inches long. The wrap angle on the small pulley is 165.6 degrees.
The speed ratio is 1 : 2.667, so if the small pulley turns at 1,750 rpm, the large one turns at 656.25 rpm. The belt moves at pi x 3 x 1,750 / 12 = 1,374.4 feet per minute, or 6.98 meters per second. The geometry was checked against the length of the two tangent lines and the arcs on each pulley, which gives the same 57.592 and 58.801.
Open and crossed belts
An open belt keeps both pulleys turning the same way, and its straight parts run on the outside of both. A crossed belt twists once and makes the pulleys turn in opposite directions. It wraps farther around each pulley, so it grips better, but it needs a center distance greater than half the sum of the diameters, and the belt must tolerate the twist.
Choosing a real belt
The calculated length is along the pitch line. Belt makers list standard lengths, so you pick the nearest size and adjust the center distance, usually with a slot or a tensioner. Allow for the belt stretching during use and for installing it without prying it over the flange. To compare belt drives with gear drives, see the Gear Ratio Calculator, and for the power a belt must carry, the Horsepower From Torque Calculator.
More motion and materials tools
For related motion and materials tools, try the Terminal Velocity Calculator, Buoyancy and Hydrostatic Pressure Calculator and Spring Oscillation Calculator.
Assumptions and limits
- The pulley diameters are the pitch diameters, where the belt actually rides.
- The belt has no stretch and no slip.
- The large pulley is at least as big as the small one, and the shafts are parallel.
- Belt thickness and tensioner travel are not included.
- The diameters and the center distance in the form are examples to replace with your own.
Frequently asked questions
How good is the common approximation?
It is very close when the pulleys are small compared with the center distance, as in the example. It gets worse when the pulleys are large for the distance, so the exact formula is shown next to it.
What is the wrap angle for?
It is the angle of belt in contact with the small pulley. A small wrap angle gives less grip, so a design needs a wrap that suits the belt type. Check the minimum in the guide from the belt maker.
Why does the page say a crossed belt needs wider spacing?
A crossed belt only works when the shafts are far enough apart for the two straight sections to cross. The condition is that the center distance exceeds half the sum of the diameters.
Does this work for a chain or a timing belt?
Only roughly. Chains and timing belts have a pitch and a number of links or teeth, so their length is a whole number of pitches. Use the calculated length as a starting point and round to the nearest size.