Timing Belt Length & Center Distance Calculator
Timing belt length and teeth from pitch, pulley teeth and center distance, or center distance from a belt. Pitch diameter, wrap, teeth in mesh, speed.
Formulas (Gates)
- Pitch diameter PD = teeth z × pitch p ÷ π
- Belt length L = p × belt teeth zb (measured on the pitch line; a whole number of pitches)
- Two equal pulleys: L = 2C + π·d
- Small pulley wrap angle θ1 = 2·arccos((d2 − d1) ÷ 2C)
- L = 2C·sin(θ1/2) + θ1·d1/2 + (2π − θ1)·d2/2 (solved numerically for C)
- Approximation: L ≈ 2C + 1.57(D + d) + (D − d)² ÷ 4C, C ≈ (Y + √(Y² − 2(D − d)²)) ÷ 4, Y = L − π(D + d)/2
- Belt speed v (m/s) = π × PD (mm) × rpm ÷ 60,000
Center distance with two equal pulleys (1:1, mm)
| Pitch / teeth | PD mm | 100T | 120T | 150T | 200T | 250T | 300T |
|---|---|---|---|---|---|---|---|
| 5 mm / 20T | 31.8 | 200.0 | 250.0 | 325.0 | 450.0 | 575.0 | 700.0 |
| 8 mm / 30T | 76.4 | 280.0 | 360.0 | 480.0 | 680.0 | 880.0 | 1080.0 |
| 5 mm / 40T | 63.7 | 150.0 | 200.0 | 275.0 | 400.0 | 525.0 | 650.0 |
At 1:1, C = (L − π·PD) ÷ 2 directly. For different pulley sizes use the calculator above (eq. (9)).
Gates manual examples (checked by tests to give the same values)
- 8 mm pitch, 72 and 144 teeth (PD 7.218 and 14.437 in), 2400 mm (94.49 in) belt → 30.02 in centers (762.5 mm).
- 14 mm pitch, 36 and 72 teeth (PD 6.316 and 12.632 in), 2310 mm (90.94 in) belt → 30.42 in.
- Belt speed: 7.218 in × 1160 rpm ÷ 3.82 = 2191.9 ft/min (about 11.1 m/s). 3.82 is 12/π rounded.
- Four rows (52 cells) of the 5 mm pitch drive selection table (p.12) were calculated: 48 match to the last digit, and the other 4 differ by 0.01 in because the calculated value sits on a rounding boundary.
How to read it
- Belt length is a whole number of pitches, so you cannot hit an arbitrary center distance exactly. Pick one of the two nearest belts and set the pulleys at its center distance, leaving room for tensioning.
- Gates' approximation (1.57 and (D − d)²/4C) drifts from the exact formula as the pulley size difference grows. This calculator solves the exact formula numerically and shows the approximation for comparison.
- Gates' rating tables assume at least six teeth in mesh on the small pulley. With fewer, ratings must be reduced, so check the wrap angle too.
- The speed ratio equals the tooth ratio (no slip). Check the output speed and belt speed together.
FAQ
What is the timing belt length formula?
For two equal pulleys, L = 2C + π·PD. For different sizes, L = 2C·sin(θ/2) + θ·d1/2 + (2π − θ)·d2/2 with θ = 2·arccos((d2 − d1)/2C); the common approximation is 2C + 1.57(D + d) + (D − d)²/4C.
How do I get the pitch diameter?
Teeth × pitch ÷ π. An 8 mm pitch, 72-tooth pulley is 72 × 8 ÷ π = 183.3 mm (7.218 in). The outside diameter is slightly smaller.
How do I find the center distance from a belt?
Enter the belt teeth or length under «Belt → center distance». The exact formula has no closed form, so it is solved numerically; the approximation is shown too.
Does it work for GT2 belts?
Yes, the formula only needs the pitch. Two 20-tooth GT2 (2 mm pitch) pulleys at 100 mm centers need a 240 mm (120-tooth) belt.
For motor speed and torque on a linear axis, see the Ball Screw Torque & Motor Sizing Calculator. For motor encoder pulses and speed, use the Encoder Pulses ↔ RPM Calculator.
Basis
Formulas are from Gates Mectrol «Timing Belt Theory» (white paper, 2006) pp.3–4, equations (4)–(10), and the «Useful Formulas and Calculations» page (p.188) of Gates' «PowerGrip GT3 Drive Design Manual» (17195, 10/2014); examples are from the same manual pp.3–4 (selection example, belt speed) and p.12 (5 mm pitch drive selection table) (all PDFs on gates.com). Tests check that this calculator reproduces those values. The six-teeth-in-mesh basis is from the manual's Engineering §14. For stock belt lengths and pulley sizes, see your maker's catalogue.