Timing belt length calculator: GT2 belt length and center distance for two pulleys

The formula

d = z × p ÷ π                                    pitch diameter of each pulley
φ = asin((D − d) ÷ 2C)                            belt angle, in radians
L = 2C × cos φ + π × (D + d) ÷ 2 + φ × (D − d)    exact belt pitch length
n = L ÷ p                                         belt teeth

• z is a pulley’s tooth count and p is the belt pitch, 2 mm for GT2. D is the larger pitch diameter and d the smaller.
• C is the center distance, from shaft center to shaft center.
• φ is the angle each straight run of belt makes with the line between the shafts. With φ in degrees, the last term is π × φ × (D − d) ÷ 180.
• L is the pitch length. For a closed loop, n has to be a whole number.

The three terms are the two straight spans, half of each pitch circle, and the extra arc the large pulley takes from the small one. Gates’ design manual gives this exact formula, and a shorter approximation in its drive selection steps:

L ≈ 2C + 1.57 × (D + d) + (D − d)² ÷ 4C

With equal pulleys φ is zero and the formula becomes L = 2C + z × p. A loop over two 20-tooth GT2 pulleys is twice the center distance plus 40 mm.

How do you calculate timing belt length for two pulleys?

Get each pulley’s pitch diameter (teeth × pitch ÷ π), then add the straight spans (2C × cos φ), half of both pitch circles and φ × (D − d). GT2 pulleys with 20 and 60 teeth at 60 mm centers need a 202.71 mm belt, 101.36 teeth, so order 101 or 102 and adjust the center distance.

Worked example: 20 and 60 teeth at 60 mm

These are the calculator’s defaults: a 3 : 1 reduction on 2 mm GT2 belt, with both pulleys in Gates’ 2 mm sprocket tables.

1. Pitch diameters: 20 × 2 ÷ π = 12.73 mm and 60 × 2 ÷ π = 38.20 mm. Gates’ tables list the same sizes as 0.501 in and 1.504 in.
2. Belt angle: φ = asin(25.46 ÷ 120) = 0.2138 rad (12.25°).
3. Straight spans: 2 × 60 × cos φ = 117.27 mm.
4. Half of both pitch circles: π × 50.93 ÷ 2 = 80.00 mm.
5. Extra arc: 0.2138 × 25.46 = 5.45 mm.
6. Total: 117.27 + 80.00 + 5.45 = 202.71 mm, or 101.36 teeth.

Gates’ approximation gives 202.66 mm, 0.05 mm short. Either way, no belt fits at exactly 60 mm.

Rounding to a whole-tooth belt

A closed loop has a whole number of teeth, so its length comes in steps of one pitch (2 mm for GT2). The calculator takes the belt just below and just above your length and solves the center distance each one needs:

Belt	Pitch length	Center distance	Change from 60 mm
101 teeth	202 mm	59.64 mm	0.36 mm closer
102 teeth	204 mm	60.66 mm	0.66 mm farther
104 teeth	208 mm	62.70 mm	2.70 mm farther
105 teeth	210 mm	63.72 mm	3.72 mm farther

Belts aren’t stocked in every tooth count. Gates’ 2 mm list has 101 teeth (202 mm) as a stock length but no 102 or 103-tooth belt. Its next lengths up are 104 teeth (208 mm, non-stock) and 105 teeth (210 mm, stock). Choose the belt from your seller’s list first, then set the shafts to the center distance it needs.

Closed loops only

A belt clamped to a printer’s carriage is open-ended belt, which Gates sells as long-length belting for reciprocating carriage drives. It is cut to fit, not sized with this formula. This calculator is for closed loops: Z belt sync loops, belted reductions on a Z axis or an extruder, and CNC or robot joints.

How do you find the center distance for a given belt length?

Solve the length formula for C by trial: guess a center distance, work out the belt length and adjust until the two agree, the iteration Gates’ design manual describes. A 100-tooth (200 mm) GT2 belt on 20 and 60-tooth pulleys needs 58.61 mm between shafts. Gates’ closed-form estimate comes within 0.03 mm.

Belt length grows steadily with center distance, so each belt has exactly one answer, and the calculator finds it by bisection. For a check by hand, Gates gives:

K = 4L − 6.28 × (D + d)
C ≈ (K + √(K² − 32 × (D − d)²)) ÷ 16

For the 200 mm belt, K = 800 − 6.28 × 50.93 = 480.16. The root is √(480.16² − 32 × 25.46²) = 458.05, so C ≈ (480.16 + 458.05) ÷ 16 = 58.64 mm.

Center distance on stock GT2 belts

Center distances for common pulley pairs on three of Gates’ stock 2 mm belts:

Pulleys	100 teeth (200 mm)	150 teeth (300 mm)	200 teeth (400 mm)
20 and 20	80 mm	130 mm	180 mm
20 and 40	69.71 mm	119.83 mm	169.88 mm
20 and 60	58.61 mm	109.26 mm	159.49 mm
16 and 80	47.57 mm	99.92 mm	150.62 mm

The 16 and 80 pair is the 5 : 1 belt reduction Klipper’s documentation uses as its gear_ratio example. On the 200 mm belt it leaves only 5 teeth in mesh on the small pulley (see below).

Leave room to tension the belt

Give a printed mount a slot or an idler so the belt can be tensioned. Gates advises against a fixed center distance except on lightly loaded or motion transfer drives. Belt and pulley tolerances add up: tension can vary by a factor of 3 or more between a long belt on small pulleys and a short belt on large ones.

A fixed center has to hold within 0.05 to 0.08 mm, and Gates says that accuracy often takes stamped steel framework. Once the belt is on, tension it with the calculator’s free span length and the belt tension calculator.

What is timing belt pitch length?

Pitch length is a belt’s circumference measured along its pitch line, which runs through the tensile cords, not over the teeth or the back. It always equals teeth × pitch, and GT2, HTD and T belts carry it in their names: Gates’ 2MR-200 is a 200 mm, 2 mm pitch belt with 100 teeth.

Pitch is the distance between two tooth centers on that line. A pulley’s pitch circle sits just outside its teeth, so its pitch diameter is larger than its outside diameter: Gates gives 12.73 mm and 12.22 mm for a 20-tooth GT2 pulley. A tape around the outside of a belt doesn’t read the pitch length. Count the teeth and multiply by the pitch.

Pitch by belt profile

Profile	Pitch	20-tooth pitch diameter	100-tooth belt
GT2 2 mm (2GT, 2MGT)	2 mm	12.73 mm	200 mm
GT2 3 mm (3GT, 3MGT)	3 mm	19.10 mm	300 mm
HTD 3M	3 mm	19.10 mm	300 mm
HTD 5M	5 mm	31.83 mm	500 mm
T2.5	2.5 mm	15.92 mm	250 mm
T5	5 mm	31.83 mm	500 mm
MXL	2.032 mm (0.080 in)	12.94 mm	203.2 mm (8 in)
XL	5.08 mm (0.200 in)	32.34 mm	508 mm (20 in)

The length math needs only the pitch, but the belt and both pulleys must share a profile. Gates says its GT belts run in GT2 sprockets and don’t fit HTD sprockets, even though both come in 3 and 5 mm pitch.

How belts are named:

• GT2, HTD and T: pitch length in mm, then pitch, then width. Gates’ 300 3MGT 09 is 300 mm long, 3 mm pitch and 9 mm wide, so it has 100 teeth.
• XL: length in tenths of an inch. Gates’ 330 XL 025 is 33.0 in long (838.2 mm) and 0.25 in wide, with 165 teeth.
• MXL: Gates asks buyers to give the pitch and the number of teeth.

Teeth in mesh and wrap angle

Gates rates its belt drives for 6 or more teeth in mesh on the smaller pulley. Teeth in mesh is the wrap angle × the teeth ÷ 360, with the fraction dropped, because a tooth that isn’t fully engaged doesn’t count as a working tooth. Below 6, Gates multiplies the torque rating by 0.8 at 5 teeth, 0.6 at 4 and 0.4 at 3. At 2 or 1 it suggests a redesign.

The wrap on the small pulley is 180° − 2φ. In the worked example that is 155.5° on the 20-tooth pulley, and 155.5 × 20 ÷ 360 = 8.6, so 8 teeth mesh. Gates’ shortcut, 180 − 60 × (D − d) ÷ C, gives 154.5°.

The count drops with a big ratio on a short belt. The 16 and 80 pair wraps only 129.3° of the small pulley on a 200 mm belt, 5 teeth in mesh. A 300 mm belt raises that to 156.5° and 6 teeth. A longer center distance or a bigger small pulley does the same.

Pulley ratio in Klipper

For a belt reduction on a stepper, Klipper’s gear_ratio is driven teeth to driving teeth. Its documentation gives gear_ratio: 80:16 for a 16-tooth motor pulley turning an 80-tooth pulley. The calculator prints the line for your pair (60:20 for the defaults). Type the ratio into the rotation distance calculator for the full printer.cfg lines.

How to measure the inputs

1. Pulley teeth: count the grooves, or read them from the part number. Gates’ 2MR-20S-06 is a 2 mm pitch pulley with 20 grooves for a 6 mm belt.
2. Center distance: measure the gap between the two shafts with calipers and add half of each shaft’s diameter.
3. A belt you already have: count its teeth and pick “Center distance from belt length”. For a belt sold by length, type the number in its name as the pitch length.