How do I calculate a timing belt's length?
Two spans plus two arcs: L = 2C·cos φ + π(D + d)/2 + φ(D − d), with φ = asin((D − d)/2C) and D, d the pitch diameters (teeth × pitch / π). For equal pulleys it is simply 2C + teeth × pitch.
GT2 belt length calculator
Enter the two pulleys' tooth counts and how far apart their shafts are: you get the exact belt length, the closed belts you can buy nearest to it and the centre distance each one sets — or start from a belt you have and get the exact centre distance. Ratio, speed and teeth in mesh come with it, and the pulleys open in the GT2 pulley generator.
Belt pitch length
360.27 mm
The short formula gives 360.27 mm.
Belt teeth
180.14
A closed belt needs a whole number: see the stock belts below.
Ratio
1 : 2
Driven pulley: 300 rpm
Teeth in mesh (small pulley)
9
Belt wraps 175.1° of it.
Pitch Ø (mm)
12.732 / 25.465
Outside Ø 12.22 / 24.96 mm.
Belt speed
0.4 m/s
Nearest closed belts
Closed GT2 belts come in whole teeth; each sets its own centre distance.
| Belt | Teeth | Centre distance | Change |
|---|---|---|---|
| 358 mm | 179 | 148.86 | -1.14 |
| 360 mm | 180 | 149.86 | -0.14 |
| 370 mm | 185 | 154.87 | +4.87 |
| 386 mm | 193 | 162.88 | +12.88 |
Make the pulleys
Each pulley opens in the GT2 pulley generator with its tooth count filled in — set the bore and belt width there, then download STL to print or STEP.
20-tooth GT2 pulley →Outside Ø 12.22 mm.
40-tooth GT2 pulley →Outside Ø 24.96 mm.
The exact belt length — and why not the short formula
A belt round two pulleys is two straight spans and two arcs. With the pitch diameters D (large) and d (small) — tooth count × pitch / π — and the centre distance C, the spans leave the line of centres at an angle φ. Gates' design manual prints both the exact length (its formula 7) and the common short one (formula 6). This calculator uses the exact one.
Going back from a belt to a centre distance has no closed form. Gates gives an approximate formula (8) and says the exact centre distance is found by iterating between the two; this calculator solves the exact equation numerically (bisection to 10⁻¹² mm), so both directions agree to the last digit.
d = z · p / π (pitch diameter; GT2: p = 2 or 3 mm) φ = asin((D − d) / 2C) L = 2C · cos φ + π (D + d) / 2 + φ (D − d) exact (Gates formula 7, φ in radians) L ≈ 2C + 1.57 (D + d) + (D − d)² / 4C approximate (Gates formula 6) C ≈ [K + √(K² − 32 (D − d)²)] / 16, K = 4L − 6.28 (D + d) approximate (Gates formula 8) teeth in mesh = z_small · (180° − 2φ) / 360°
How far apart are they? For a 20 → 60 tooth drive at 150 mm the two lengths differ by less than a thousandth of a millimetre; at 45 mm on 16 → 60 teeth the short formula is off by more. The calculator shows both. Checked against Gates' 5 mm drive selection tables (8 rows, both directions, to the tables' 0.01 in).
Worked example: a 20-tooth motor pulley driving 60 teeth, 150 mm apart
Pitch diameters 40 / π = 12.732 mm and 120 / π = 38.197 mm; φ = asin(25.465 / 300) = 4.869°. The exact length is 381.081 mm — 190.54 teeth, so no closed belt fits exactly. The nearest belts sold are 370 mm (185 teeth), which sets the shafts 144.438 mm apart, and 386 mm (193 teeth), 152.468 mm — pick the one your adjustment can reach, or move the shafts to suit.
Ratio 1 : 3: a motor at 600 rpm turns the 60-tooth pulley at 200 rpm, and the belt moves 20 × 2 mm × 600 / 60 = 0.4 m/s. The belt wraps 170.26° of the small pulley, so 9 of its 20 teeth are in mesh — Gates' manual rates belts on at least 6.
Closed GT2 belt lengths
A closed belt is a whole number of teeth, so its length is always an even number of millimetres — but not every even length is made. These are the lengths two sellers list: Ondrives' closed 2GT belts (the manufacturer's catalogue, 4 and 6 mm wide) and the 6 mm closed loops VXB sells for 3D printers. The calculator picks its nearest belts from this list.
Longer than these, or a length nobody stocks: use an open belt cut to length and clamped, as printer X/Y axes do.
| Belt length | Teeth | Listed by |
|---|---|---|
| 110 mm | 55 | VXB |
| 112 mm | 56 | VXB |
| 122 mm | 61 | VXB |
| 154 mm | 77 | Ondrives |
| 158 mm | 79 | VXB |
| 176 mm | 88 | Ondrives |
| 182 mm | 91 | Ondrives |
| 188 mm | 94 | Ondrives |
| 200 mm | 100 | VXB |
| 210 mm | 105 | Ondrives |
| 258 mm | 129 | Ondrives |
| 280 mm | 140 | VXB |
| 282 mm | 141 | Ondrives |
| 290 mm | 145 | Ondrives |
| 300 mm | 150 | VXB |
| 302 mm | 151 | Ondrives |
| 320 mm | 160 | Ondrives |
| 322 mm | 161 | Ondrives |
| 332 mm | 166 | Ondrives |
| 340 mm | 170 | Ondrives |
| 346 mm | 173 | Ondrives |
| 358 mm | 179 | Ondrives |
| 360 mm | 180 | Ondrives |
| 370 mm | 185 | Ondrives |
| 386 mm | 193 | Ondrives |
| 400 mm | 200 | VXB |
| 420 mm | 210 | Ondrives |
| 426 mm | 213 | Ondrives |
| 430 mm | 215 | Ondrives |
| 444 mm | 222 | Ondrives |
| 448 mm | 224 | Ondrives |
| 460 mm | 230 | Ondrives |
| 478 mm | 239 | Ondrives |
| 484 mm | 242 | Ondrives |
| 486 mm | 243 | Ondrives |
| 506 mm | 253 | Ondrives |
| 544 mm | 272 | Ondrives |
| 600 mm | 300 | Ondrives |
| 610 mm | 305 | VXB |
| 616 mm | 308 | Ondrives |
| 630 mm | 315 | Ondrives |
| 852 mm | 426 | VXB |
Leave room to fit and tension the belt
Gates' manual (section II-9) gives the centre distance a drive needs to move: less to slip a belt on, more to tension it over its life. For belts of 125–250 mm, 0.8 mm each way; 250–500 mm, 1.0 mm in and 0.8 mm out; 500–1000 mm, 1.8 mm in and 0.8 out — more again if you have to go over flanged pulleys.
A fixed centre distance is for light motion only: Gates wants it held within 0.05–0.08 mm, and recommends an idler on the slack side when the shafts can't move. On a 3D printer that is the usual belt tensioner.
FAQ
What GT2 belt for 20 and 40 teeth at 100 mm?
260.41 mm — 130.2 teeth. The nearest closed belts sold are 258 mm (129 teeth), which puts the shafts 98.79 mm apart, and 280 mm (140 teeth), 109.82 mm.
What centre distance does a 200 mm GT2 belt give on two 20-tooth pulleys?
Exactly 80 mm: with equal pulleys C = (L − z·p) / 2 = (200 − 40) / 2.
Closed loop or open belt?
A closed (endless) belt has no joint and comes only in whole teeth, so the centre distance follows the belt. An open belt is cut to length and clamped — what 3D-printer X/Y axes use, where the belt's ends are fixed to the carriage.
How tight should the belt be?
Ondrives: tight enough that the slack side stays taut under the highest load — any sag or flap means too little. Gates measures it by the force to deflect the span 1/64 in per inch of span.
What steps/mm does a GT2 pulley give?
One turn moves teeth × 2 mm of belt: 40 mm on 20 teeth. A 200-step motor at 16 microsteps makes 3200 steps a turn: 3200 / 40 = 80 steps/mm.
Sources
Every number on this page comes from these documents (the version or the date we read them is given).
- Gates PowerGrip GT2 Belt Drive Design Manual (The Gates Rubber Company, © 2000), Engineering section II-2, I-11 – I-13, II-7, II-9 — exact and approximate length formulas (6, 7), the approximate centre-distance formula (8), teeth in mesh, installation and tensioning allowances; drive selection tables used as test data
- Ondrives, Timing Belts and Pulleys — Technical (updated March 2024) — pitch circle d0 = z·t/π, belt length for i = 1 and i ≠ 1, belt speed, teeth in mesh, pre-tension
- Ondrives, Timing Pulleys & Belts — GT catalogue (product information updated April 2017; distributor copy, Amironic) — closed 2GT belts 2GT-4 / 2GT-6, pp. 7–8: the manufacturer's stock lengths
- VXB, GT2 closed loop timing belt 2GT 6 mm (read 2026-10-20) — the closed loops sold for 3D printers: 110 … 852 mm (a retailer)
- Forgent3D GT2 pulley generator — evidence pack (Misumi, Ondrives, SDP/SI, Adafruit outside diameters) — pitch line differential 0.254 mm (2 mm) / 0.381 mm (3 mm) for the outside diameter