Gears · Standard reference geometry
Spur Gear Dimensions Calculator
Use this spur gear calculator to find pitch, outside, root, and base diameters from module or diametral pitch, tooth count, and pressure angle.
Reference calculator #005
Enter standard spur-gear geometry
Inputs stay in your browser. Values are normalized to canonical units before calculation.
Calculated output
Results
- Outside diameter, dₐ
- Root diameter, df
- Base diameter, db
- Circular pitch, p
- Reference tooth thickness, s
- Addendum, hₐ
- Dedendum, hf
- Whole depth, h
- Simplified undercut reference
Valid standard full-depth reference geometry
- One external standard full-depth involute spur gear uses zero profile shift.
- The basic rack has addendum coefficient 1.0 and dedendum coefficient 1.25.
- Pressure angle is the transverse reference pressure angle; module is on the same transverse plane.
- Dimensions are nominal reference geometry; backlash, tolerances, cutter details, modifications, strength, and manufacturing allowances are not evaluated.
Calculation engine: spur-gear-dimensions/1.0.0
Spur gear dimension formulas
The calculator models one external, standard full-depth involute spur gear with zero profile shift. Module, tooth count, and transverse pressure angle are treated as nominal reference-geometry inputs.
| Symbol | Meaning | V1 relationship |
|---|---|---|
m |
Module | input in mm/tooth, or 25.4 / Pᵈ |
z |
Tooth count | positive whole number |
α |
Transverse pressure angle | input in degrees |
d |
Reference pitch diameter | mz |
dₐ |
Outside or tip diameter | m(z + 2) |
df |
Root diameter | m(z − 2.5) |
db |
Base diameter | d cos α |
p |
Circular pitch | πm |
s |
Reference tooth thickness | πm / 2 |
hₐ, hf, h |
Addendum, dedendum, whole depth | m, 1.25m, 2.25m |
Worked module example
For m = 2 mm, z = 20, and α = 20°:
- Pitch diameter:
d = 2 × 20 = 40 mm. - Outside diameter:
dₐ = 2 × (20 + 2) = 44 mm. - Root diameter:
df = 2 × (20 − 2.5) = 35 mm. - Base diameter:
db = 40 cos 20° = 37.587705 mm. - Circular pitch:
p = π × 2 = 6.283185 mm. - Reference tooth thickness:
s = p / 2 = 3.141593 mm.
The calculation engine retains floating-point precision. Display rounding is applied only after the domain result is produced.
Worked diametral-pitch example
For Pᵈ = 20 teeth/in, z = 20, and α = 20°, the equivalent module is 1.27 mm. The results are a 1 in pitch diameter, 1.1 in outside diameter, 0.875 in root diameter, and approximately 0.939693 in base diameter.
How to use the spur gear calculator
Choose the pitch system used by the drawing or catalogue. Enter module in millimetres per tooth or diametral pitch in teeth per inch, then enter the gear tooth count and transverse pressure angle. The URL records only these inputs, so a shared link reproduces the calculation without creating duplicate content URLs.
Use the diagram to compare the four calculated reference circles. It deliberately does not draw involute teeth or cutter fillets: a visually plausible tooth outline would imply geometry that this V1 calculator does not compute.
Simplified undercut reference
The engineering warning compares the tooth count with zmin = 2 / sin²α. At 20°, this gives about 17.10 teeth, so a 17-tooth, zero-shift input receives a warning while the dimensional result remains available. This is a preliminary reference, not a manufacturing acceptance rule.
Engineering scope and limitations
These are nominal basic-rack dimensions, not finished inspection dimensions. The calculator does not evaluate:
- profile shift, stub teeth, nonstandard addendum or dedendum, protuberance, tip relief, or crowning;
- helical, internal, bevel, worm, rack, or asymmetric gears;
- cutter-generated root fillets, undercut extent, backlash, tolerances, quality grade, or metrology;
- bending strength, contact stress, material, heat treatment, lubrication, efficiency, noise, life, or load capacity;
- mating-gear compatibility, operating center distance, contact ratio, or final manufacturing acceptance.
Confirm the drawing convention, basic rack, tooling, and applicable design standard before releasing a gear for manufacture.
Frequently asked questions
How do you calculate spur gear pitch diameter?
Multiply module by the whole-number tooth count: d = mz. In a diametral-pitch system, the equivalent relationship is d = z/Pᵈ when d is in inches and Pᵈ is in teeth per inch.
How do you calculate the outside diameter of a standard spur gear?
For the zero-profile-shift, standard full-depth geometry used here, outside diameter is dₐ = m(z + 2). Different basic racks, profile shifts, tip modifications, or finished dimensions require different inputs.
How do you calculate spur gear root diameter?
This V1 model uses a dedendum of 1.25m, giving df = m(z - 2.5). Actual root form and finished root diameter depend on the cutter, fillet, profile shift, and manufacturing method.
Can I enter diametral pitch instead of module?
Yes. Select Diametral pitch and enter teeth per inch. The calculator converts it to canonical module internally with m = 25.4/Pᵈ, then displays length results in inches.
Does the undercut warning prove that a gear is safe or unsafe?
No. The warning uses the simplified zero-profile-shift reference zmin = 2/sin²α. Cutter geometry, basic rack, profile shift, helix angle, and design standards can change the limit, so verify the actual tooth system.
References and review status
Reviewed . References provide independent formula checks and terminology context; they do not imply endorsement or standards certification.
- KHK Gears — Calculation of Gear Dimensions — Public technical reference used to cross-check standard full-depth spur-gear diameter and tooth-proportion formulas.
- ISO 21771-1:2024 — Cylindrical involute gears and gear pairs, Part 1 — Official terminology and geometry context. ISO currently marks this edition for revision and lists a replacement draft under development.