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`"belt length" = pi * ( "R" + "r" ) + 2* asin( ( "R" - "r" ) / "L" )*( "R" - "r" ) + 2 * sqrt( "L" ^2 - ( "R" - "r" )^2 )`

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The **Two Pulley Belt Length** equation computes the length of a belt that goes around two pulleys, where the pulleys may be of unequal radii.

**INSTRUCTIONS:** Choose your preferrred units and enter the following:

- (
**R**) The radius of the larger pulley - (
**r**) The radius of the smaller pulley - (
**L**) The distance between the axles

**Belt Length:** The calculator returns the belt length in inches. However, this can be automatically converted to other length units via the pull-down menu.

See the video explanation of this equation's derivation.

From the picture, see that the following describe the component parts of the solution:

`C_R` = half the larger pulley circumference = `pi / 2 * 2R`, where 2R = larger pulley diameter

`C_r` = half the smaller pulley circumference = `pi / 2 * 2r`, where 2r = larger pulley diameter

**u** = segment of larger circumference between half circle and tangent point of pulley = `theta/(2pi) * (pi * 2R)`

**v** = segment of smaller circumference between half circle and tangent point of pulley = `theta/(2pi) * (pi * 2r)`

b = length of belt between tangent point on two pulleys = `sqrt( L^2 - (R-r)^2 )`

`L * sin( theta ) = R - r` ` =>` `theta = asin( (R-r)/L )`

Belt Length = (`C_R + 2u`) + (`C_r - 2v`) + 2b

= (`piR + 2* thetaR`) + (`pir - 2* thetar`) +2b

= `pi ( R+r) + 2theta* (R-r) + 2b`

= `pi ( R+r) + 2* asin( (R-r) / L) * (R-r) + 2*sqrt( L^2 - (R-r)^2)`