BIZOMAT MAKEOS · Power Transmission
Crossed Belt and Three-Pulley Belt Length
How long is the belt for this crossed drive or for a belt around three pulleys?
Belt length for a crossed (reversing) two-pulley drive, with the open-belt length for comparison, or for a belt wrapped around the outside of three pulleys at given positions, with wrap angles.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Drive | choice | Crossed two-pulley drive, or open belt around three pulleys (pulley 1 at the origin). |
| Pulley 1 diameter | mm | Pitch diameter of pulley 1. |
| Pulley 2 diameter | mm | Pitch diameter of pulley 2. |
| Pulley 3 diameter (three-pulley) | mm | Pitch diameter of pulley 3 (three-pulley drive). |
| Centre distance (crossed) | mm | Centre distance of the crossed drive. |
| Pulley positions (three-pulley) | ||
| Pulley 2 X | mm | X position of pulley 2 centre (pulley 1 is at 0, 0). |
| Pulley 2 Y | mm | Y position of pulley 2 centre. |
| Pulley 3 X | mm | X position of pulley 3 centre. |
| Pulley 3 Y | mm | Y position of pulley 3 centre. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Drive
- What it is
- The drive layout: a crossed belt between two pulleys (which reverses the direction of rotation), or an open belt running around the outside of three pulleys.
- Units and format
- Choose one of 2 options from the list: Crossed belt, two pulleys; Open belt around three pulleys.
- Example
- The form opens with “Open belt around three pulleys” selected as a worked example. Choose the option that matches your case.
- Get it right
- The crossed drive uses the centre distance; the three-pulley drive uses the pulley positions instead.
What this means: Pulley 1 diameter
- What it is
- The pitch diameter of pulley 1, measured at the belt’s pitch line rather than the outside of the rim. In the three-pulley drive its centre is at the origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the pitch diameter from the pulley catalogue, not the outside diameter.
What this means: Pulley 2 diameter
- What it is
- The pitch diameter of pulley 2, measured at the belt’s pitch line rather than the outside of the rim.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the pitch diameter from the pulley catalogue, not the outside diameter.
What this means: Pulley 3 diameter (three-pulley)
- What it is
- The pitch diameter of the third pulley in a three-pulley drive, such as an idler or second driven pulley. Only the three-pulley drive uses it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- All three pulleys must sit outside the belt loop; a back-bent idler is not covered.
What this means: Centre distance (crossed)
- What it is
- The distance between the centres of the two pulleys in the crossed drive. Only the crossed drive uses it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 600 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must be larger than the average of the two diameters.
Pulley positions (three-pulley)
What this means: Pulley 2 X
- What it is
- The horizontal position of pulley 2’s centre, measured from pulley 1’s centre at the origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 600 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only in the three-pulley drive.
What this means: Pulley 2 Y
- What it is
- The vertical position of pulley 2’s centre, measured from pulley 1’s centre at the origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only in the three-pulley drive.
What this means: Pulley 3 X
- What it is
- The horizontal position of pulley 3’s centre, measured from pulley 1’s centre at the origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 300 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The three centres must not lie in a straight line.
What this means: Pulley 3 Y
- What it is
- The vertical position of pulley 3’s centre, measured from pulley 1’s centre at the origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 250 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The three centres must not lie in a straight line.
Method and formulas
- Crossed: L = 2√(C² − ((D + d)/2)²) + (D + d)/2 × (π + 2 arcsin((D + d) ÷ 2C)).
- Three pulleys: tangent length between pulleys i and j = √(dᵢⱼ² − (rᵢ − rⱼ)²); the belt’s outward normal on each span is found from cos(angle) = (rᵢ − rⱼ) ÷ dᵢⱼ.
- Arc on each pulley = radius × wrap angle between its incoming and outgoing spans; L = Σ spans + Σ arcs.
- Example: the default three-pulley drive needs a 1,810.5 mm belt.
Limitations
- Pitch-line lengths; the belt maker’s length designation (inside or outside) may differ.
- All three pulleys must be on the outside of the belt loop (no back-bent idler).
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Order belts for reversing drives
- Find belt length around an idler or third pulley
- Check wrap on each pulley
Who it is for
- Design engineers
- Maintenance engineers
What you get
- Belt length
- Wrap angle on each pulley
- Open-belt length (crossed drive comparison)
Interpreting the result
- Keep at least about 120° of wrap on the driving pulley.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2473–2490. Topic reference; no source tables or recommended settings are reproduced.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
Open the Crossed Belt and Three-Pulley Belt Length calculator