AtlasCalcsUtility Calculation Suite
August 2026 12 min read

Spur Gear Calculator

Free spur gear calculator. Compute pitch diameter, addendum, dedendum, whole depth, circular pitch, tooth thickness and center distance from module or diametral pitch, with an undercut (minimum-teeth) check.

Spur Gear Calculator

Spur Gear Inputs

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

Tooth size

Gear 1 Teeth z₁
Gear 2 Teeth z₂
Pressure Angle

Sets undercut limit

Auto-calculate instantly

Center Distance

80mm

d₁ = 40 mm, d₂ = 120 mm.

Geometry is valid; no undercut
Pitch Dia d₁

40mm

Pitch Dia d₂

120mm

Module

2mm

Diametral Pitch

12.70DP

Addendum

2mm

Dedendum

2.500mm

Whole Depth

4.500mm

Circular Pitch

6.283mm

Pitch Dia d₁

40mm

m·z₁

Pitch Dia d₂

120mm

m·z₂

Module

2mm

Tooth size

Diametral Pitch

12.70DP

25.4/m

Addendum

2mm

a = m

Dedendum

2.500mm

b = 1.25m

Whole Depth

4.500mm

2.25m

Circular Pitch

6.283mm

π·m

Result Summary

Module2 mm
DP12.70
z₁ / z₂20 / 60 T
Pitch d₁40 mm
Pitch d₂120 mm
Center Distance80 mm
Whole Depth4.500 mm
Tooth Thickness3.142 mm
Undercut RiskNo
Estimated Result — Uses standard full-depth involute proportions: addendum = m, dedendum = 1.25 m, whole depth = 2.25 m, circular pitch = π m. The undercut check compares each gear to the minimum teeth for the chosen pressure angle (≈17 at 20°, 13 at 25°). Standard systems assume a standard (non-shifted) tooth; profile shift, backlash trimming and AGMA strength rating are not included.

Interactive Diagram

Standard spur tooth: addendum above and dedendum below the pitch circle, whole depth 2.25·m.

Visualizations

Spur Gear Dimensions (mm)

40mmd₁
120mmd₂
2mma
2.5mmb
4.5mmh
80mmC

Pinion Teeth vs Undercut Limit

z₁ = 20, min 17118%

Formula

A spur gear's geometry follows directly from the module (tooth size) and tooth count. Pitch diameter scales the gear; addendum/dedendum set tooth height; center distance spaces the pair; circular pitch sets tooth spacing. The undercut limit guards against weak roots.

d = m · z

Alternative Form

a = m , b = 1.25m , h = 2.25m , C = m(z₁+z₂)/2 , p = π·m

Variables

SymbolNameUnitDescription
mModulemmPitch diameter per tooth
zTeeth—Number of teeth
dPitch diametermmm·z
a, bAddendum, dedendummma = m, b = 1.25 m
hWhole depthmm2.25 m
CCenter distancemmm(z₁+z₂)/2
pCircular pitchmmπ·m

Explanation

Once you fix the module (from a standard series) and the tooth counts (from the ratio and center distance), every other dimension is determined by the standard tooth proportions. Keep the pinion at or above the undercut limit so the involute profile is not cut away at the root; if you cannot, use profile shift or a higher pressure angle. The center distance must be held within tight tolerance for the teeth to mesh correctly.

Derivation

Module is defined as m = d/z, so d = m·z. Standard full-depth teeth set addendum = 1·m and dedendum = 1.25·m, giving whole depth 2.25·m and clearance 0.25·m. Center distance for a mating pair is the sum of pitch radii: C = (d₁+d₂)/2 = m(z₁+z₂)/2. Circular pitch p = π·m; tooth thickness at the pitch circle is half of that for equal tooth/space.

Step-by-Step Calculation

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

Step 1

Determine the module (or diametral pitch).

m = 25.4 / DP
DP:12.70
Intermediate: m = 2 mm
Result: m = 2 mm
Step 2

Compute the pitch diameters.

d = m·z
z₁:20
z₂:60
Intermediate: d₁ = 2×20, d₂ = 2×60
Result: d₁ = 40 mm, d₂ = 120 mm
Step 3

Find addendum, dedendum and whole depth.

a = m , b = 1.25m , h = 2.25m
m:2 mm
Intermediate: a = 2, b = 2.5
Result: h = 4.500 mm
Step 4

Compute circular pitch, tooth thickness and base pitch.

p = π·m , t = p/2 , p_b = p·cos(α)
α:20°
Intermediate: p = 6.283
Result: t = 3.142 mm, p_b = 5.904 mm
Step 5

Set the center distance.

C = m(z₁ + z₂)/2
z₁:20
z₂:60
Intermediate: C = 2×(20+60)/2
Result: 80 mm
Final Answer
80 mm

Worked Examples

See how the calculator works with different scenarios.

Scenario

Module 2 mm, 20/40 tooth pair.

Inputs

m:2 mm
z₁:20
z₂:40
α:20°

Calculation

d₁ = 40 mm, d₂ = 80 mm; a = 2, b = 2.5, h = 4.5 mm; C = (40+80)/2 = 60 mm; p = 6.283 mm, t = 3.142 mm

Result

C = 60 mm, whole depth 4.5 mm

Interpretation

A clean standard pair with comfortable tooth proportions.

Understanding Your Result

Your Result: 80 mm

Typical range: Standard modules: 1–10 mm common. Your m = 2 mm is in the common range.

What This Means

For m = 2 mm, z₁ = 20, z₂ = 60, the pitch diameters are 40 mm and 120 mm, center distance 80 mm, whole depth 4.500 mm. Undercut risk: no.

Why It Matters

Tooth geometry fixes how two gears fit and share load. Wrong center distance or undercut pinions cause noise, wear and early failure. Getting addendum/dedendum and circular pitch right is the first step before any strength calculation.

Professional Interpretation

Confirm the pair on a standard module, hold center distance to 80 mm, and if z₁ < 17 apply profile shift or raise α. Then run AGMA/ISO 6336 for bending and pitting before manufacture.

Undercut

Both gears ≥ 17 teeth — no undercut.

Tooth height

a = 2 mm, b = 2.500 mm, h = 4.500 mm.

Spacing

p = 6.283 mm, t = 3.142 mm.

Recommendations

  • •No undercut risk; geometry is sound.
  • •Choose a standard module (1, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10).
  • •Hold center distance within a few hundredths of mm for proper mesh.
  • •Add small backlash by trimming tooth thickness, not by changing pitch.
  • •Follow with AGMA/ISO tooth-strength rating for the load.

Educational Content

Introduction

The spur gear is the workhorse of power transmission: straight teeth, parallel shafts, simple to make. Before you can rate a gear for strength you must define its geometry — and that flows almost entirely from two numbers: the module (tooth size) and the tooth count. This calculator returns every standard spur-gear dimension from those inputs and flags undercut risk.

Concept Explanation

Module m = pitch diameter ÷ teeth ties everything together: d = m·z. Standard full-depth teeth set the addendum to one module and the dedendum to 1.25 modules (whole depth 2.25 m, clearance 0.25 m). Circular pitch p = π m sets tooth spacing, and half of it is the nominal tooth thickness. Center distance C = m(z₁+z₂)/2 spaces the pair. The undercut limit — about 17 teeth at 20° pressure angle — is the minimum count before the cutter removes root material and weakens the tooth.

Applications

▸Sizing spur-gear pairs from ratio and center distance
▸Selecting a standard module and tooth counts
▸Checking undercut before manufacture
▸Teaching involute tooth geometry
▸Generating inspection dimensions
▸Feeding AGMA/ISO strength calculations

Benefits

  • •Returns all standard tooth dimensions from module + teeth
  • •Converts module ↔ diametral pitch
  • •Flags undercut (minimum teeth) automatically
  • •SI / imperial consistent
  • •Diagrams the tooth profile

Limitations

  • •Standard (non-shifted) tooth only
  • •No profile shift / backlash trimming
  • •No bending or pitting strength rating
  • •Single pair, parallel shafts
  • •No tolerance/layout sizing

Best Practices

  • •Pick a standard module from the ISO series
  • •Keep pinion ≥ undercut minimum, or use profile shift
  • •Hold center distance to tight tolerance
  • •Use 20° (or 25°) pressure angle for strength
  • •Follow with AGMA/ISO 6336 strength checks

Common Misconceptions

Myth: Any tooth count works.
Truth: Below the undercut limit the root is cut away and the tooth weakens.
Myth: Module and DP are interchangeable.
Truth: They are inverse systems (m = 25.4/DP) and not mixable.
Myth: Center distance is flexible.
Truth: It must equal m(z₁+z₂)/2 or the mesh is wrong.
Myth: Tooth thickness equals circular pitch.
Truth: It is half the circular pitch for a standard gear.

Practical Tips

  • •Center distance 80 mm; whole depth 4.500 mm.
  • •No undercut risk with current teeth.
  • •Keep center distance within a few hundredths of mm.
  • •Add backlash by trimming tooth thickness, not pitch.
  • •Run AGMA/ISO strength rating for the real load.

Common Mistakes

Mixing module and DP gears
Pick one system; convert with m = 25.4/DP.
Under-cut pinion
Add teeth, raise pressure angle to 25°, or apply profile shift.
Wrong center distance
Set shafts exactly to m(z₁+z₂)/2.
Ignoring backlash
Trim tooth thickness slightly; do not alter the module.
Skipping strength check
Geometry is only half — rate bending/pitting per AGMA/ISO.

Assumptions

This calculator makes the following assumptions to provide estimates:

Full-depth standard tooth: a = m, b = 1.25 m, h = 2.25 m.
No profile shift: Standard (unshifted) tooth assumed.
Valid teeth: Positive integer-ish tooth counts entered.
Undercut limit: Approximate min teeth per pressure angle.
No strength rating: AGMA/ISO capacity is separate.

Limitations

Be aware of these limitations when using this calculator:

No shift/backlash: Profile shift and backlash trimming excluded.
No strength: Bending/pitting needs AGMA/ISO 6336.
Single pair: One mating pair, parallel shafts.
Ideal geometry: Manufacturing tolerances ignored.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a spur gear?
A spur gear has straight teeth parallel to the axis. It transmits power between parallel shafts with 97–99% efficiency but can be noisier than a helical gear at speed.
What is gear module?
Module m = pitch diameter ÷ teeth. It is the fundamental metric gear size; standard values run from 0.3 to 50 mm (common 1, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10).
What is diametral pitch?
Diametral pitch DP is the imperial equivalent: teeth per inch. DP = T / pitch diameter (in). Convert: m = 25.4 / DP. A 2 mm module ≈ 12.7 DP.
How is pitch diameter calculated?
d = m × z (mm). For a 2 mm module and 30 teeth, d = 60 mm. It is the theoretical contact diameter used for center distance and ratio.
What is addendum and dedendum?
Addendum a = m (top of tooth above pitch circle). Dedendum b = 1.25 m (root below pitch circle, gives clearance). Whole depth h = a + b = 2.25 m.
What is the pressure angle?
Angle between the line of action and the direction of motion. Standard: 14.5° (obsolete), 20° (common), 25° (stronger, more bearing load).
What is circular pitch?
p = π × m = π × d / z — the pitch-circle distance between corresponding points on adjacent teeth. Used for cutter setup and tooth thickness.
What is tooth thickness?
Standard thickness at the pitch circle is half the circular pitch: t = p / 2 = π m / 2, giving equal tooth and space width (backlash is then added by trimming).
How do I calculate center distance?
C = (d₁ + d₂) / 2 = m (z₁ + z₂) / 2. It is the shaft-center spacing and must be held tightly for proper mesh.
What is backlash in spur gears?
Clearance between mating flanks that prevents jamming from thermal growth and tolerances. Precision gears: 0.02–0.10 mm. Too much adds noise and positioning error.
What is the minimum number of teeth?
To avoid undercutting: 17–18 teeth at 20°, 14–15 at 25°, 18 at 14.5°. Below this the cutter removes involute material at the root, weakening the tooth.
What is undercutting?
Material removed at the tooth root during cutting when teeth are too few for the pressure angle, weakening the tooth and lowering contact ratio.
What is contact ratio?
Average number of teeth in contact; ≥1.2 recommended (1.4–1.8 is quieter). Higher values smooth load sharing.
How do I calculate spur gear speed (pitch-line velocity)?
v = π d n / 60000 m/s (d in mm, n in rpm). Above ~25 m/s, precision grinding and special lube are needed.
What are standard modules?
ISO first-choice: 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10 mm. Pick from these for off-the-shelf tooling.
What is the difference between module and DP?
Module is mm/tooth (metric); DP is teeth/inch (imperial). They are not interchangeable — m = 25.4 / DP.
How do I calculate gear strength?
Use the Lewis formula or AGMA. Lewis: F_t = σ · b · m · Y, where σ is allowable stress, b face width, Y the Lewis form factor (≈0.484 − 2.87/z for 20°).
What is the Lewis form factor?
Y accounts for tooth shape and stress concentration: for 20° involute, Y ≈ 0.484 − 2.87/z (z ≥ 17). It falls as teeth decrease, raising bending stress.
What materials are used for spur gears?
Through-hardened steel (1045, 4140) general; case-carburized (8620, 4320) heavy; stainless for corrosion; bronze/brass light; nylon/acetal quiet, light duty.
How are spur gears manufactured?
Hobbing (production), shaping (internal/shoulders), milling (prototypes), grinding (precision), injection molding (plastic), powder metallurgy (mid-volume).
What is the AGMA quality number?
Q5–Q15 define tolerances; higher is tighter. Q5–Q6 commercial, Q7–Q8 precision, Q9–Q10 high, Q11+ ultra (special grinding).
What is gear crowning?
Slight convex tooth modification (0.005–0.025 mm/mm face) to avoid edge loading from misalignment, improving load distribution and quieting operation.
How does face width affect capacity?
Capacity scales with face width b, but b/d = 0.3–1.0 is typical; very wide faces risk uneven load from shaft deflection.
What is gear lubrication?
Grease (low speed, enclosed), splash (oil bath, moderate), forced circulation (high speed/load), oil mist (very high speed) — all form an EHL film.
What causes gear noise?
Tooth impact, pitch/profile errors, runout, vibration, bearing and resonance noise. Fix with higher contact ratio, helical teeth, grinding, damping.
What is the efficiency of spur gears?
97–99% per mesh. Losses: tooth sliding 1–2%, churn 0.5–1%, bearings 0.5–1%. Box efficiency = product of meshes.
How do I select spur gear size?
Driven by power, speed, center distance, material, life, noise and cost. Rough module from torque: m ≈ (T/50)^(1/3) mm.
What are internal spur gears?
Teeth on the inside of a ring meshing with an external pinion; compact coaxial drives with high contact ratio — used in planetary sets.
How do I measure spur gears?
Tooth count, pitch diameter (over-pins), module (gear-tooth vernier), thickness (gear-tooth mic), runout (dial), hardness, surface finish.
What is root fillet radius?
Curved root transition; larger radius cuts stress concentration and boosts bending strength (~0.38 m for hobbed gears).
How does heat treatment affect gears?
Through-hardening (HRC 30–40) general; case-carburizing (HRC 58–62 surface) heavy; nitriding (HRC 50–60) low-distortion; induction selective.
How do I prevent gear failure?
Right material/heat treat, lubrication, alignment, safety factors, inspection for pitting/wear, vibration monitoring, stay within ratings.
What is the difference between spur and helical gears?
Spur teeth are straight and axial-thrust-free but noisier; helical teeth are angled, quieter, higher capacity, but produce axial thrust.
Can I mix metric and imperial gears?
No — module and DP profiles differ and are not interchangeable. Convert carefully: DP = 25.4 / m.
What is whole depth of a spur gear tooth?
Total radial tooth height h = 2.25 m for standard full-depth teeth (addendum m + dedendum 1.25 m).
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Glossary

Technical terms used in mortgage calculations explained in plain language.

Module (m)
Pitch diameter per tooth (mm/tooth); tooth size.
Diametral Pitch (DP)
Teeth per inch; imperial tooth size.
Pitch Diameter (d)
d = m·z; theoretical contact diameter.
Addendum (a)
a = m; tooth height above pitch circle.
Dedendum (b)
b = 1.25 m; tooth depth below pitch circle.
Whole Depth (h)
h = 2.25 m = a + b.
Circular Pitch (p)
p = π·m; pitch distance between teeth.
Base Pitch
p·cos(α); distance between load paths.
Undercut
Root material loss when teeth are too few for α.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

AGMA 2001/2101ISO 6336ISO 54ISO 1328DIN 867

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Page Information

Last Updated:

August 2026

Version:

2.1.0

License:

Free to use for educational and professional purposes.