Use this calculator to find the theoretical finished outside diameter of a standard, unmodified external spur gear from tooth count and either module or diametral pitch. It also reports the reference diameter, nominal root diameter, and nominal whole tooth depth.
Calculate standard spur-gear dimensions
Deliberate calculation mode
Choose the tooth-size system used on the drawing. Results update only when you select Calculate or press Enter in an input.
Nominal geometry results
Scope: these are reference-geometry values, not stock allowance, tolerances, cutter-generated root form, mesh acceptance, or load capacity.
Report a possible calculation error
The report is prepared only after you select the button. Nothing is transmitted automatically.
How to use the gear blank calculator
- Select metric module or imperial diametral pitch to match the gear drawing.
- Enter the module or diametral-pitch value and an integer number of teeth.
- Select Calculate gear dimensions to create a current result.
- Check the declared assumptions before using the theoretical tip, reference, root, or whole-depth values in a drawing or process plan.
Live gear-circle visualizer
This schematic uses the same calculation engine as the standard mode. It shows how the tip, reference, and nominal root circles move as module and tooth count change. It is not a tooth-profile drawing and is not to scale for cutter fillets or undercut.
Text alternative: at module 2 mm and 20 teeth, the tip diameter is 44 mm, reference diameter is 40 mm, nominal root diameter is 35 mm, and nominal whole tooth depth is 4.5 mm.
FIRGELLI Automations — Interactive Engineering Calculators
What does “gear blank diameter” mean here?
Here it means the theoretical finished tip-circle diameter of the specified tooth geometry. It is the circle through the tooth tips after the gear form is produced. A raw turning blank may require process-specific stock, tolerance, finishing, coating, or heat-treatment allowance; this calculator does not add any of those values.
The reference diameter is the geometric reference used to define module or diametral pitch. “Pitch diameter” is common shop language for this value, but operating pitch diameter can differ in a modified mesh, so the results label uses reference diameter.
Module and diametral pitch without unit ambiguity
Transverse module m is reference diameter in millimetres divided by tooth count, so its unit is mm/tooth. Transverse diametral pitch P is tooth count divided by reference diameter in inches, so its unit is teeth/in. They describe the same tooth size in reciprocal forms:
m [mm/tooth] = 25.4 / P [teeth/in]
For a spur gear, normal and transverse sections coincide. Do not apply this direct conversion to a helical-gear value unless you first establish whether the drawing specifies normal or transverse module/pitch and account for helix angle.
Equations and symbols
| Symbol | Quantity | Metric relationship | Imperial relationship |
|---|---|---|---|
| z | integer tooth count | dimensionless | |
| m, P | module; diametral pitch | m in mm/tooth | P in teeth/in; m = 25.4/P |
| d | reference diameter | d = zm | d = z/P |
| da | theoretical tip diameter | da = m(z + 2) | da = (z + 2)/P |
| df | nominal root diameter | df = m(z − 2.5) | df = (z − 2.5)/P |
| h | nominal whole depth | h = 2.25m | h = 2.25/P |
Dimensional check: z and all numerical coefficients are dimensionless, while m is a length per tooth; therefore each metric result is a length. Since 1/P has units inches per tooth, each imperial result is inches.
Worked metric and imperial examples
Metric: module 2 mm, 20 teeth
da = 2(20 + 2) = 44 mm; d = 2(20) = 40 mm; df = 2(20 − 2.5) = 35 mm; h = 2.25(2) = 4.5 mm. KHK publishes the same reference example.
Imperial: 10 diametral pitch, 30 teeth
da = (30 + 2)/10 = 3.2 in; d = 30/10 = 3 in; df = (30 − 2.5)/10 = 2.75 in; h = 2.25/10 = 0.225 in. The equivalent module is 25.4/10 = 2.54 mm/tooth.
Shop-floor check: module 3 mm, 24 teeth
The theoretical tip diameter is 3(24 + 2) = 78 mm; reference diameter is 3(24) = 72 mm; nominal root diameter is 3(24 − 2.5) = 64.5 mm; and nominal whole depth is 2.25(3) = 6.75 mm. The 78 mm result is not automatically the raw stock order diameter. Add only the allowance and tolerance required by the drawing, material, heat-treatment route, workholding, and finishing process.
How should the results be interpreted?
- Theoretical tip diameter is the finished circle through tooth tips for the declared standard geometry. It is not automatically the raw stock order size.
- Reference diameter sets tooth spacing and size. It is not a physical edge to measure directly.
- Nominal root diameter is the standard reference-circle result. The manufactured root may differ because a cutter generates a fillet and may undercut a low-tooth-count gear.
- Nominal whole depth is radial tip-to-root depth from the 1.00m addendum plus 1.25m dedendum. It is not a cutting-depth instruction for every tool or process.
When is this relationship useful?
Use it for preliminary geometry, drawing review, stock-envelope discussion, or an independent arithmetic check when the gear specification explicitly matches the declared unshifted external spur-gear system. It is also useful for checking whether a module and tooth count are internally consistent with a stated theoretical outside diameter.
When should it not be used?
Do not use these equations as-is for internal, helical, bevel, worm, hypoid, rack, stub-tooth, non-involute, profile-shifted, or addendum-modified gears. The calculator does not select a cutter; check undercut or interference; calculate backlash, center-distance modification, tooth thickness, strength, life, efficiency, noise, or quality grade; or establish manufacturing allowances and tolerances.
A positive numerical root diameter is only an algebraic circle under this model. It does not prove that the generated root is physically identical, free of undercut, or manufacturable. Low tooth counts commonly require closer tooth-generation review, but this page does not assert a universal minimum because the relevant limit depends on pressure angle, profile shift, cutter, and mesh requirements.
Common mistakes and how to prevent them
- Calling raw stock OD the formula result. Keep theoretical finished tip diameter separate from process allowance and tolerance.
- Mixing module and diametral pitch. Module grows with tooth size; diametral pitch decreases. Keep their units visible.
- Using a fractional tooth count. A complete gear has an integer count; the calculator rejects fractions.
- Ignoring profile shift. A nonzero addendum modification changes tip and root geometry, so use the drawing’s specified method.
- Treating the root circle as the exact cutter path. Confirm cutter tip radius, protuberance, clearance, and undercut separately.
What should be checked next?
Confirm the drawing’s pressure angle, tooth-depth system, profile-shift coefficient, module or diametral-pitch plane, cutter, finished-OD tolerance, process allowance, and heat-treatment sequence. Then evaluate mesh geometry, interference, backlash, tooth strength, contact stress, lubrication, shaft/bearing loads, and applicable safety factors as separate engineering tasks.
Manufacturing and inspection considerations
This calculator defines nominal reference geometry; it does not define a complete manufacturing blank. The process plan should separately specify rough-turning stock, workholding allowance, finish-machining sequence, coating or heat-treatment compensation, and the finished outside-diameter tolerance. Those values depend on the drawing, material, gear quality requirement, equipment, and process capability, so a universal extra allowance or tolerance would be misleading.
Before tooth cutting, inspect the turned blank against the drawing. Typical checks include outside diameter at several angular positions, roundness, face and outside-diameter runout relative to the selected datum, bore or shaft-seat geometry, and any surface condition required by the cutting process. The acceptable limits must come from the applicable drawing and process controls, not from this calculator.
For reverse engineering, tooth count and a measured outside diameter can provide a tentative module or diametral-pitch estimate only when the gear is an unmodified external full-depth spur gear and the tooth tips are not worn, damaged, or intentionally altered. Confirm the result against pitch measurements, pressure angle, profile shift, center distance, and a recognized gear standard before treating it as the original specification.
Frequently asked questions
What is the difference between module and diametral pitch?
Module is reference diameter in millimetres divided by tooth count, while diametral pitch is tooth count divided by reference diameter in inches. For the same transverse tooth size, m = 25.4/P. A larger module means larger teeth; a larger diametral-pitch value means smaller teeth.
Why does the outside-diameter equation add 2 to the tooth count?
In the declared standard full-depth system, addendum is 1.00 module above the reference circle. Diameter spans both sides of the gear, so the theoretical tip diameter adds two addenda: da = m(z + 2). A profile shift or modified addendum changes that relationship.
How accurate must the gear blank diameter be?
There is no universal tolerance that is correct for every gear. This calculator returns nominal geometry only. The finished tolerance and any rough-stock allowance must come from the drawing, gear quality requirement, material, heat-treatment route, cutting method, finishing sequence, and demonstrated process capability.
Can I use this calculator for non-standard gear profiles?
No. The equations apply only to the declared unmodified external 20-degree full-depth spur-gear model. Stub teeth, profile shift, nonstandard addendum or dedendum, internal gears, helical gears, bevel gears, worms, racks, and non-involute forms require the applicable geometry and standard.
How do material and heat treatment affect blank sizing?
They can affect process allowance, sequencing, distortion risk, and final inspection, but this calculator does not predict those effects. The manufacturing plan should define any compensation from measured process data, supplier guidance, the material specification, and the finished drawing instead of applying a generic extra diameter.
How does gear blank diameter relate to center distance?
Blank outside diameter does not set the nominal spacing of a standard gear pair. For two unmodified spur gears with the same module, nominal center distance is (d1 + d2)/2 = m(z1 + z2)/2. Use the reference diameters and then check profile shift, backlash, and operating geometry separately.
Video walkthrough: how to use this calculator
FIRGELLI’s walkthrough demonstrates the inputs and a worked use of the Gear Blank Diameter Calculator. The video is loaded from YouTube only after you select play.
Open the Gear Blank Diameter Calculator walkthrough on YouTube.
Engineering references
- Kohara Gear Industry Co., Ltd. (KHK), The ABC's of Gears — Basic Guide, pp. 14–15. Full-depth tooth proportions and diameter equations. Accessed 2026-07-29.
- Kohara Gear Industry Co., Ltd. (KHK), “Calculation of Gear Dimensions,” section 4.1. Standard and profile-shifted spur-gear geometry. Accessed 2026-07-29.
- University of Cambridge Department of Engineering, “Spur gears: tooth generation”. 20° full-depth rack proportions and generated root context. Accessed 2026-07-29.
- War Department, Army Air Forces, TM 1-421 Aircraft Maintenance and Repair, 1942, p. 112. Imperial diametral-pitch definitions and outside-diameter relationship. Accessed 2026-07-29. Its historical dedendum convention is not used for the root result.
This calculator is intended for education, concept evaluation, and preliminary design. Results are based on the equations and assumptions described on this page, but cannot account for every real-world load case, tolerance, material property, environmental condition, installation detail, safety factor, code, or regulatory requirement. Verify all inputs, assumptions, units, and results independently before selecting components or using the result in a real application.
Safety-critical, structural, medical, lifting, transportation, or regulated applications must be reviewed by a qualified engineer.
Need to implement these calculations?
Explore the precision-engineered motion control solutions used by top engineers.
