How far does a rack move per pinion revolution?

Travel equals the pinion’s pitch circumference: π·m·z. A module 2, 20-tooth pinion moves the rack 125.66 mm per revolution; a 20 DP, 20-tooth pinion moves it 3.142 in.

How long is a gear rack with n teeth?

L = n·π·m along the pitch line. Twenty teeth at module 2 is 125.66 mm.

Where does the pinion sit relative to the rack?

The pinion center is d/2 from the rack pitch line, where d = m·z. For a 20-tooth module 2 pinion that is 20 mm; add the pitch-line-to-back distance to locate it from the rack back.

How much force does a rack and pinion produce?

F = 2T ÷ d before losses. 5 N·m of torque on a 40 mm pitch diameter pinion pushes the rack with about 250 N.

How do I join two racks end to end?

Keep the pitch continuous across the joint, so the distance between the last tooth on one rack and the first on the next is exactly p = π·m. Ending each rack at mid-space makes that easy, and a short piece of matching rack can align the joint.

What is the smallest pinion for a rack?

At 20°, about 17 teeth avoid undercut without profile shift; at 14.5° it is about 32. Smaller pinions work with a positive shift, which also moves the pinion center out by x·m.

Can I make a laser-cut rack and pinion?

Yes, for low to moderate speed and load. Laser-cut plate parts suit prototypes, robots, fixtures, and light machinery; critical, high-speed, or high-load drives need engineering review of material, thickness, hub attachment, alignment, lubrication, and wear.

Power transmission calculator

Gear Rack Generator

Generate a gear rack and mating pinion from module or diametral pitch, with linear pitch, rack length, travel per revolution, and linear force, then export STEP, DXF, or SVG.

Design a laser-cut gear rack

Set the pitch, rack length, and pinion, then place the mounting holes and choose a bore and plate.

Units

Presets

Pitch system

Module

Pressure angle

Rack teeth

Back heightPitch line to back edge

Backlash per partTooth thinning

Pinion teeth

Pinion torqueOptional

N·m

Rack mounting holes

A row of holes evenly spaced along the back of the rack.

Holes

Hole diameter

From back edge

Pinion bore, holes, and plate

Bore

Bore diameter

Mounting holes

Lightening holes

Fabworks plate material

Plate thicknessMin. feature 0.050 in · 1.27 mm

Rack dimensions

Linear pitch: 6.283 mm
Rack length Ends stop mid-gap so racks butt together on pitch: 125.664 mm
Overall rack height Back edge to tooth tips: 17 mm
Travel per pinion turn: 125.664 mm
Pinion pitch diameter: 40 mm
Pinion outside diameter: 44 mm
Pinion center to rack pitch line: 20 mm
Pinion center to rack back: 35 mm
Contact ratio: 1.769

Mesh preview

The rack has straight flanks at the pressure angle and rounded roots. The pinion's involute teeth are generated by rolling that same rack profile through each tooth space. The dashed circle is the pinion's pitch circle, which rolls along the rack's pitch line.

True generated tooth form

Download the manufacturing files

100% in-browser

STEP solids are built and checked in your browser. DXF and SVG drawings use lines and true arcs at full scale. Nothing is uploaded until you order.

Gear rack
125.7 mm × 17.0 mm
Pinion
⌀ 43.6 mm

Laser cut it at Fabworks

We generate the STEP files, add them to a new quote, and open it in a new tab where you can review material, finish, and quantity.

Quick answer

How this gear rack calculator works

A gear rack’s linear pitch is p = π·m, so a rack of n teeth is n·π·m long and a pinion of z teeth moves it π·m·z per revolution; with m = 2 and a 20-tooth pinion that is 125.66 mm per turn. This generator draws the rack and its mating pinion with mounting holes and gives center distance and linear force. Files are generated in the browser and download as STEP, DXF, or SVG, or go straight into a Fabworks quote.

How to use the calculator

1Choose module or diametral pitch and the pressure angle, matching the pinion you will use.
2Enter the number of rack teeth, the back height, and backlash.
3Enter the mating pinion teeth to get travel per revolution and the pinion center position.
4Add mounting holes along the rack, and optionally enter pinion torque to get linear force.
5Pick a Fabworks material and thickness and clear any minimum-feature warnings.
6Download the rack and pinion as STEP, DXF, or SVG, or send them to a Fabworks quote.

Key formulas

Gear rack formulas

Linear pitch: p = π·m = π ÷ DP; Spacing between rack teeth along the pitch line.
Rack length: L = n·π·m; n is the number of rack teeth.
Travel per pinion revolution: s = π·m·z = π·d; z is the pinion tooth count and d its pitch diameter.
Pinion center to rack pitch line: h = d/2 = m·z/2; Add x·m if the pinion has profile shift.
Pinion center to rack back: H = d/2 + distance from pitch line to rack back; Use this to locate the pinion shaft from the rack’s mounting surface.
Linear force from torque: F = 2T ÷ d; T is pinion torque and d its pitch diameter. 5 N·m on a 40 mm pinion gives 250 N.
Linear speed: v = π·d·n; n is pinion speed in revolutions per unit time.
Rack tooth height: addendum = m, dedendum = 1.25m; Measured above and below the pitch line.

Step by step

How to draw a gear rack profile

1Draw the pitch line and mark tooth centers every p = π·m.
2Make each tooth π·m/2 thick on the pitch line, less half the backlash on each flank.
3Draw both flanks as straight lines inclined at the pressure angle α from the tooth centerline; a rack’s involute is a straight line.
4Draw the tip line m above the pitch line and the root line 1.25m below it.
5Round the root corners with a fillet of about 0.38m.
6Repeat for n teeth, then add the back height and mounting holes.
7Place the pinion center d/2 from the pitch line (plus x·m for a shifted pinion).

Worked example

Module 2 rack with a 20-tooth pinion

Inputs: m = 2 mm, 20° pressure angle, 20 rack teeth, 20-tooth pinion.
Result: Linear pitch 6.283 mm, rack length 125.66 mm, travel 125.66 mm per pinion revolution, pinion center 20 mm from the rack pitch line.

p = π × 2 = 6.283 mm. Length = 20 × 6.283, and travel = π × 2 × 20. The pinion’s pitch diameter is 40 mm, so its center sits 20 mm above the pitch line.

Common questions

What to know before using the result

How far does a rack move per pinion revolution?: Travel equals the pinion’s pitch circumference: π·m·z. A module 2, 20-tooth pinion moves the rack 125.66 mm per revolution; a 20 DP, 20-tooth pinion moves it 3.142 in.
How long is a gear rack with n teeth?: L = n·π·m along the pitch line. Twenty teeth at module 2 is 125.66 mm.
Where does the pinion sit relative to the rack?: The pinion center is d/2 from the rack pitch line, where d = m·z. For a 20-tooth module 2 pinion that is 20 mm; add the pitch-line-to-back distance to locate it from the rack back.
How much force does a rack and pinion produce?: F = 2T ÷ d before losses. 5 N·m of torque on a 40 mm pitch diameter pinion pushes the rack with about 250 N.
How do I join two racks end to end?: Keep the pitch continuous across the joint, so the distance between the last tooth on one rack and the first on the next is exactly p = π·m. Ending each rack at mid-space makes that easy, and a short piece of matching rack can align the joint.
What is the smallest pinion for a rack?: At 20°, about 17 teeth avoid undercut without profile shift; at 14.5° it is about 32. Smaller pinions work with a positive shift, which also moves the pinion center out by x·m.
Can I make a laser-cut rack and pinion?: Yes, for low to moderate speed and load. Laser-cut plate parts suit prototypes, robots, fixtures, and light machinery; critical, high-speed, or high-load drives need engineering review of material, thickness, hub attachment, alignment, lubrication, and wear.

Reference

Rack travel per pinion revolution (mm)

Travel = π·m·z. Linear pitch p = π·m is the rack tooth spacing.

Module	Linear pitch	12-tooth pinion	15-tooth	20-tooth	30-tooth
0.5	1.571	18.85	23.56	31.42	47.12
1	3.142	37.70	47.12	62.83	94.25
1.25	3.927	47.12	58.90	78.54	117.81
1.5	4.712	56.55	70.69	94.25	141.37
2	6.283	75.40	94.25	125.66	188.50
2.5	7.854	94.25	117.81	157.08	235.62
3	9.425	113.10	141.37	188.50	282.74
4	12.566	150.80	188.50	251.33	376.99
5	15.708	188.50	235.62	314.16	471.24
6	18.850	226.19	282.74	376.99	565.49

Rack travel per pinion revolution, diametral pitch (in)

Travel = π·z ÷ DP. Linear pitch is π ÷ DP.

Diametral pitch	Linear pitch	12-tooth pinion	15-tooth	20-tooth	30-tooth
32	0.0982	1.178	1.473	1.963	2.945
24	0.1309	1.571	1.963	2.618	3.927
20	0.1571	1.885	2.356	3.142	4.712
16	0.1963	2.356	2.945	3.927	5.890
12	0.2618	3.142	3.927	5.236	7.854

Formula

p = π·m · L = n·π·m · travel per turn = π·m·z

Linear pitch p is π times the module, so a rack of n teeth is n·π·m long. A pinion of z teeth moves the rack π·m·z per revolution and pushes it with F = 2T ÷ d.

Engineering references

Assumptions and limits

Rack teeth use straight flanks at the pressure angle, with addendum 1.0m and dedendum 1.25m.
The pinion center sits d/2 from the rack pitch line with no profile shift.
Linear force is ideal and excludes friction and mesh losses.
Backlash is applied by thinning teeth by the entered allowance.
Laser-cut racks suit low to moderate speed and load; critical drives need engineering review.

Educational estimate

Use this result for learning and early design exploration. Verify safety-critical or production decisions with the governing standard, material data, real tooling, and a qualified engineer.

Keep designing

Fabworks resources

Related calculators

Order Sheet Metal Parts

Upload a 2D DXF or 3D STEP file for an instant laser cutting quote. Quote in seconds, order in minutes, receive parts in days.

Browse Files

or drag and drop

STEP / DXF up to 24MB

Your files are safe, secure, and retain all intellectual rights.