AtlasCalcsUtility Calculation Suite
August 2026 12 min read

Rack and Pinion Calculator

Free rack and pinion calculator. Compute pinion pitch diameter, linear travel per revolution, linear velocity and rack thrust from module, pinion teeth, speed and torque. SI and imperial (module/DP), with diagram.

Rack and Pinion Calculator

Rack and Pinion Inputs

Unit System
SI — Module (mm) Imperial — DP
Module m (mm)

Tooth size

Pinion Teeth z

≥17 at 20° PA

Input Speed (rpm)

Optional

Input Torque (N·m)

Optional

Torque Unit
N·m lbf·in
Auto-calculate instantly

Linear Travel / Rev

125.66mm

Pinion pitch diameter d = 40 mm.

Valid geometry; add speed and torque for linear velocity and rack force.
Pinion d

40mm

Circular Pitch

6.283mm

Outside Dia

44mm

Linear Velocity

2.094m/s

Rack Force

500N

Pinion d

40mm

m·z

Circular Pitch

6.283mm

π·m

Outside Dia

44mm

d + 2m

Linear Velocity

2.094m/s

π·d·n/60

Rack Force

500N

2T/d

Result Summary

Module2 mm
Pinion Teeth20
Pitch Diameter40 mm
Outside Diameter44 mm
Circular Pitch6.283 mm
Travel / Rev125.66 mm
Linear Velocity2.094 m/s
Rack Force500 N
Estimated Result — Uses exact rotary-to-linear relations: d = m·z, travel/rev = π·d, circular pitch p = π·m, outside diameter d_a = d+2m, velocity v = π·d·n/(60·1000) (m/s, d in mm) and rack thrust F = 2T/d (N, d in m). It does not rate tooth strength (AGMA 2001 / ISO 6336), nor account for backlash, stiffness or wear.

Interactive Diagram

The rack advances by the pinion circumference per revolution; tooth spacing equals π·m.

Visualizations

Pinion Dimensions

40mmPitch d
44mmOutside d_a
125.7mmTravel/rev

Formula

A rack and pinion is a rotary-to-linear transformer. The pinion pitch diameter is d = m·z exactly like a spur gear. One full revolution advances the rack by the pinion pitch circumference π·d, so travel per turn is fixed by pinion size. Linear velocity follows from input rpm, and the rack thrust follows from input torque by F = 2T/d.

travel / rev = π · d

Alternative Form

v = π·d·n/(60·1000) (m/s) , F = 2T/d (N)

Variables

SymbolNameUnitDescription
mModulemmTooth size (pinion = rack)
zPinion teeth—Number of pinion teeth
dPitch diametermmm·z
pCircular pitchmmπ·m (rack spacing)
nSpeedrpmPinion speed
TTorqueN·mPinion input torque
vVelocitym/sLinear speed
FRack forceN2T/d thrust

Explanation

Because the rack is just a gear of infinite radius, each pinion turn pulls the rack forward by exactly the pitch circumference π·d. Doubling z (and d) doubles travel per turn but halves the force for the same torque — a direct speed/force trade. Unlike a worm, the motion is reversible and efficient (~95–98%), but it has backlash and is not self-locking.

Derivation

d = m·z. Per revolution the contact point travels one circumference, so linear advance s = π·d. With rotational speed n (rpm), angular speed ω = 2πn/60 rad/s and v = ω·(d/2)/1000 = π·d·n/(60·1000) m/s (d in mm). Tangential force at the pitch circle: T = F·(d/2) → F = 2T/d (d in m). Circular pitch p = π·m and rack addendum = m, dedendum = 1.25m.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Pinion pitch diameter.

d = m·z
m:2 mm
z:20
Intermediate: d = 2×20
Result: 40 mm
Step 2

Linear travel per revolution.

travel = π·d
d:40 mm
Intermediate: travel = π×40
Result: 125.66 mm
Step 3

Circular pitch.

p = π·m
m:2 mm
Intermediate: p = π×2
Result: 6.283 mm
Step 4

Linear velocity from input speed.

v = π·d·n / (60·1000)
n:1000 rpm
Intermediate: v = π·40·1000/60000
Result: 2.094 m/s
Step 5

Rack thrust from input torque.

F = 2·T / d
T:10 N·m
Intermediate: F = 2·T/(d/1000)
Result: 500 N
Final Answer
125.66 mm

Worked Examples

See how the calculator works with different scenarios.

Scenario

m = 2 mm, z = 20, n = 1000 rpm, T = 10 N·m.

Inputs

m:2 mm
z:20
n:1000 rpm
T:10 N·m

Calculation

d = 40 mm; travel/rev = π·40 = 125.7 mm; v = π·40·1000/60000 = 2.09 m/s; F = 2·10/0.04 = 500 N

Result

125.7 mm/rev, 500 N

Interpretation

A standard 40 mm pinion converts 10 N·m into 500 N of linear thrust and 2.09 m/s.

Understanding Your Result

Your Result: 125.66 mm

Typical range: Travel/rev 125.7 mm; velocity 2.094 m/s; force 500 N. Keep pinion teeth ≥ 17 (20° PA) to avoid undercut.

What This Means

This pinion (d = 40 mm, 20 teeth) advances the rack 125.66 mm per turn, moving at 2.094 m/s and pushing 500 N.

Why It Matters

Rack and pinion turns motor rotation into precise linear travel. Knowing travel per turn, velocity and force lets you size the motor, choose pinion teeth for the needed resolution, and confirm stiffness against the required thrust.

Professional Interpretation

Rate tooth bending/pitting per AGMA 2001 / ISO 6336 using m and z. Account for backlash (anti-backlash pinion) and support stiffness. Force ≈ 500 N at the pinion; velocity ≈ 2.094 m/s.

Travel per rev

125.66 mm

Circular pitch

6.283 mm

Outside diameter

44 mm

Recommendations

  • •Keep pinion teeth ≥ 17 (20° PA) to avoid undercut.
  • •Use an anti-backlash pinion for precision positioning.
  • •It is not self-locking — add a brake to hold position.
  • •Rate tooth strength per AGMA/ISO separately.
  • •Match rack support stiffness to the required force.

Educational Content

Introduction

A rack and pinion is the simplest way to turn rotation into straight-line motion. This calculator gives the pinion diameter, the exact linear travel per revolution, the velocity from your input speed, and the rack thrust from your input torque — with a module/DP unit toggle.

Concept Explanation

The pinion is a spur gear: d = m·z. Meshed with the rack, one revolution moves the rack by its pitch circumference π·d. Linear velocity v = π·d·n/60 (mm/s), and the rack thrust from torque is F = 2T/d. The rack tooth pitch equals the pinion circular pitch p = π·m; addendum is m, dedendum 1.25m.

Applications

▸Automotive steering racks
▸CNC and 3D-printer linear axes
▸Lifts and hoists
▸Sliding gates and doors
▸Robotic linear joints
▸Aircraft and instrument controls

Benefits

  • •Exact rotary-to-linear conversion
  • •High stiffness and speed
  • •Reversible (unlike a worm)
  • •Simple, low-cost, easy to align
  • •Module/DP consistent

Limitations

  • •Backlash → lost motion
  • •Not self-locking
  • •Need support for long racks
  • •No strength rating here
  • •Stiffness depends on mounting

Best Practices

  • •Use an anti-backlash pinion for accuracy
  • •Keep teeth ≥ 17 at 20° PA
  • •Add a brake if position must hold
  • •Rate teeth per AGMA/ISO
  • •Support the rack to limit deflection

Common Misconceptions

Myth: Travel per turn depends on speed.
Truth: It is fixed by π·d; speed sets velocity, not travel/rev.
Myth: It is self-locking.
Truth: It is reversible; needs a brake to hold.
Myth: More teeth always better.
Truth: More teeth = more travel/rev but less force for given torque.
Myth: Backlash is unavoidable.
Truth: Anti-backlash pinions remove most of it.

Practical Tips

  • •Travel/rev 125.7 mm.
  • •Rack force 500 N.
  • •Keep pinion teeth ≥ 17 to avoid undercut.
  • •Not self-locking — add a brake to hold.
  • •Use anti-backlash pinion for precision.

Common Mistakes

Confusing travel/rev with velocity
Travel/rev = π·d (fixed); velocity adds rpm.
Underestimating backlash
Use anti-backlash pinion or closed-loop control.
Assuming self-locking
Add a brake; it is reversible.
Under-min teeth
Keep z ≥ 17 at 20° PA (or profile shift).
Skipping strength
Rate per AGMA 2001 / ISO 6336.

Assumptions

This calculator makes the following assumptions to provide estimates:

Conjugate mesh: Pinion and rack share module m.
No backlash model: Ideal zero-backlash geometry.
Efficiency: Not applied (≈95–98% spur mesh).
Unit toggle: Module/DP and N·m/lbf·in consistent.
No rating: Strength is separate.

Limitations

Be aware of these limitations when using this calculator:

No strength rating: AGMA/ISO capacity not computed.
Ignores backlash: Real meshes have lost motion.
Support not modeled: Deflection depends on mounting.
Ideal friction: Efficiency not applied.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a rack and pinion?
A rack is a straight bar with gear teeth; a pinion is a round gear. Meshed, they convert rotation to linear motion (and back). Used in steering, CNC, lifts and actuators.
How far does the rack move per pinion turn?
Exactly the pinion circumference: travel/rev = π·d = π·m·z. One full turn advances the rack by the pitch circumference of the pinion.
How is pinion diameter calculated?
d = m·z, where m is the module and z the number of pinion teeth (same as a spur gear).
What is the rack tooth pitch?
Circular pitch p = π·m, the centre-to-centre spacing of the rack teeth — equal to the pinion circular pitch at the pitch line.
What is the linear velocity?
v = (π·d/1000)·n/60 m/s, where d is in mm and n in rpm; equivalently v = π·d·n/60 mm/s or π·d·n/60·(1/25.4) in/s.
How is rack force calculated?
Tangential force F = 2·T/d, with T the pinion torque and d the pitch diameter (use metres for N). It is the thrust the rack pushes with.
What is the pinion outside diameter?
d_a = d + 2·m (addendum = m). The rack addendum is m and dedendum 1.25m.
What is the mechanical advantage?
It is a rotary-to-linear transformer: torque becomes linear force by F = 2T/d, while speed becomes velocity by v = ω·(d/2).
Why use rack and pinion?
Simple, reversible, precise linear motion with high stiffness; ideal for positioning where a screw would be long or heavy.
What are the disadvantages?
Backlash causes lost motion; it is not self-locking; long racks need support; efficiency is like a spur mesh (~95–98%).
What is backlash?
Clearance between pinion and rack flanks; reduces accuracy. Use anti-backlash pinions (two preloaded halves) for precision.
What materials are used?
Pinion: hardened steel; rack: steel (ground) or hardened for precision, sometimes plastic/nylon for light duty.
What is the difference from a leadscrew?
A leadscrew converts rotation via a thread (can be self-locking, lower speed); rack-and-pinion is faster, reversible, lower friction, but has backlash.
What is a rack module?
Same m as the pinion so the teeth are conjugate; circular pitch π·m ties the tooth spacing.
How do I choose pinion teeth?
More teeth → larger d → more linear travel per turn and lower force for a given torque, but a bigger pinion. Fewer teeth risk undercut (keep ≥ 17 at 20° PA).
What is the minimum pinion teeth?
About 17 at 20° pressure angle to avoid undercut; 14 at 25° or with profile shift.
What is the efficiency?
Around 95–98% (like a spur mesh); much higher than a worm, but not self-locking.
Can it be back-driven?
Yes — pushing the rack turns the pinion (unlike a self-locking worm). A brake/lock is needed to hold position.
What is the AGMA standard?
AGMA 201.02 / 2001 for spur (rack) geometry and 2001-B88 for rating; ISO 6336 for strength.
What lubrication is needed?
Grease or oil (like a spur gear); for long travel, a central lube or sealed pinion. Dirty environments need wipers.
What is the stroke length?
Rack length sets max travel; pinion rotation maps linearly, so travel = π·d·(θ/360).
How accurate is it?
Accuracy is set by backlash, tooth accuracy grade and support stiffness; ground racks reach a few microns over a metre.
What are typical applications?
Car steering, CNC axes, 3D printers, lifts, robotic joints, sliding gates, aircraft controls.
What is the force if I know the motor torque?
F = 2·T/d. A 10 N·m motor on a 30 mm pinion gives 667 N of rack thrust.
What speed if I know rpm?
v = π·d·n/60 (mm/s). A 30 mm pinion at 1000 rpm moves 1570 mm/s = 1.57 m/s.
Can the ratio be changed?
No fixed ratio like gears; linear motion scales with pinion size. Changing d changes travel/rev and force together.
What is a helical rack?
A rack with angled teeth meshing a helical pinion — quieter, higher load, but with axial thrust to absorb.
What is the cost?
Pinion $10–$200; ground steel rack $50–$500/m depending on accuracy; long precision tracks cost more.
What failures occur?
Wear, pitting, backlash growth, support deflection and misalignment. Keep teeth lubricated and aligned.
How do I reduce backlash?
Use an anti-backlash (spring-loaded dual) pinion, tighten mesh, or close-loop control with an encoder.
Does the calculator include force and speed?
Yes — given input speed and torque it returns linear velocity v = π·d·n/60 and rack thrust F = 2T/d.
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Glossary

Technical terms used in mortgage calculations explained in plain language.

Travel / Rev
Linear advance per pinion turn = π·d.
Circular Pitch (p)
π·m; rack tooth spacing.
Pitch Diameter (d)
m·z of the pinion.
Outside Diameter
d + 2m (addendum = m).
Rack Force (F)
2T/d; thrust on the rack.
Backlash
Flank clearance; lost motion.
Anti-backlash
Spring-loaded dual pinion.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

AGMA 2001-B88ISO 6336DIN 867

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Page Information

Last Updated:

August 2026

Version:

2.1.0

License:

Free to use for educational and professional purposes.