How this calculation works
A belt follows straight tangent spans and arcs around pulley pitch circles. Its path length is therefore slightly different from a simple pair of centre distances plus half the two circumferences when pulley sizes differ.
Exact path = two tangent spans + contact arcs; open-belt tangent angle = asin((large radius − small radius)/centres)A worked example
Two equal 100 mm pitch-diameter pulleys at 500 mm centres in an open arrangement have two 500 mm straight spans and a combined 100 pi mm of contact arc. The path is 1000 plus 100 pi, or about 1314.159 mm. For equal pulleys the open-path approximation is exact; changing one diameter makes the tangent angle and approximation error informative.
Good to know
Ideal circular pitch radii and a perfectly flexible path; no belt thickness, stretch, catalogue stock number, tension or power rating. Confirm manufacturer pitch-length definitions and installation take-up requirements. A calculated geometric path is not a recommended purchased belt length.
What this model answers
A belt follows straight tangent spans and arcs around pulley pitch circles. Its path length is therefore slightly different from a simple pair of centre distances plus half the two circumferences when pulley sizes differ.
This page calculates the ideal tangent-and-arc geometry and also shows the common long-centre approximation. Their difference helps reveal when an approximation is becoming less suitable, without pretending the ideal path is already a catalogue belt selection.
The Pulley RPM ratio and shaft power calculator uses the pitch-diameter ratio to estimate driven speed. Path length sets geometric fit; it does not establish shaft speed, slip or transmitted power.
Measure the right inputs
Enter pitch diameters rather than outside pulley diameters unless the component specification defines them as equivalent. Belt thickness and pulley groove geometry can make those dimensions different.
The centres must separate the two pitch circles. Open paths use external tangents; crossed paths use internal tangents and a different arc geometry. A crossed path here is a geometric option, not a claim that a particular belt construction permits crossing.
Read the result in context
Small-pulley contact angle is shown because the wrap can change substantially with size difference and centre spacing. The calculator does not convert that angle into traction or power capacity.
The approximation-error metric is signed. It compares the stated long-centre formula with the exact ideal path, not with a measured installed belt. Stretch, take-up and manufacturing tolerances are separate from mathematical approximation error.
Work through the numbers
Two equal 100 mm pitch-diameter pulleys at 500 mm centres in an open arrangement have two 500 mm straight spans and a combined 100 pi mm of contact arc.
The path is 1000 plus 100 pi, or about 1314.159 mm. For equal pulleys the open-path approximation is exact; changing one diameter makes the tangent angle and approximation error informative.
Where the model stops
Ideal circular pitch radii and a perfectly flexible path; no belt thickness, stretch, catalogue stock number, tension or power rating.
Confirm manufacturer pitch-length definitions and installation take-up requirements. A calculated geometric path is not a recommended purchased belt length.
A few questions, answered
Can I enter pulley outside diameter?
Only if you have established that it matches the required pitch diameter. Groove geometry often makes them different.
Does crossed mode approve crossing a timing belt?
No. It calculates ideal internal-tangent geometry only, without checking a belt construction or operating suitability.