About Spur Gear Dimensions Calculator (Module & Diametral Pitch)
The spur gear dimensions calculator turns a tooth count and a tooth size — diametral pitch in the inch world, module in the metric world — into the full blank and tooth geometry: pitch, outside, root, and base circle diameters, addendum, dedendum, whole depth, clearance, and circular pitch and thickness. Every proportion comes from the printed standards tables in Machinery's Handbook: ANSI B6.1-1968 (R1974) full-depth proportions for diametral pitch, and the Handbook's DIN 867 module table for metric.
The two systems are exact inverses through 25.4 (module = 25.4 ÷ DP — the Handbook's own rule), but their printed dedendums differ: ANSI's preferred inch dedendum is 1.25/P, while the Handbook's DIN 867 metric table prints 1.157 × module with the 0.157m clearance American cutter makers standardized on (a 1.167m one-sixth-clearance variant is also printed; many modern ISO-based designs run deeper roots — your cutter or standard governs). Enter a mating tooth count and the calculator adds the center distance and ratio. Standard full-depth external gears at 20° pressure angle only — profile shift, backlash allowance, and internal gears are design work beyond a proportions table, and tooth counts below the Handbook's 18-tooth minimum get an undercut warning.
How It Works
- Pick the system. Diametral pitch (teeth per inch of pitch diameter) drives inch gears — bigger P means smaller teeth; module (millimetres of pitch diameter per tooth) drives metric gears — bigger m means bigger teeth. Toggling converts at the exact inverse 25.4/P and clamps to the caps (10 DP ↔ 2.54 module, the Handbook's own example).
- Enter the tooth count. Pitch diameter follows definitionally (D = N/P or m·N), and the standard proportions hang off the tooth size: inch gears take addendum 1/P and dedendum 1.25/P per ANSI B6.1 Table 2; metric gears take addendum m and dedendum 1.157m per the printed DIN 867 form.
- Read the blank dimensions: outside diameter (N+2)/P or m(N+2) — what you turn the blank to — root diameter, whole depth (2.25/P or 2.157m) — what the cutter must reach — and the base circle D·cos 20°, the circle the involute actually unwinds from.
- Add the mating gear's tooth count for the center distance — (N₁+N₂)/2P or m(N₁+N₂)/2, the number the housing bores must hold — and the ratio. Gears only mesh at equal pitch/module and pressure angle.
- Watch the undercut flag: below 18 teeth (the Handbook's printed minimum for 20° full depth), a generated pinion's flank gets undercut at the root, weakening it and shortening contact. Real small-pinion designs enlarge the pinion (profile shift) — a modification this proportions calculator deliberately leaves to gear-design references.
Worked Example
A 24-tooth, 8-DP pinion: pitch diameter 24/8 = 3.0000 in, outside diameter 26/8 = 3.2500 in — turn the blank to that — root diameter (24−2.5)/8 = 2.6875 in, addendum 0.1250, dedendum 0.1563, whole depth 0.2813 in of cutter travel, circular pitch π/8 = 0.3927 in, and a base circle of 3 × cos 20° = 2.8191 in. Meshing with a 48-tooth gear: center distance (24+48)/(2×8) = 4.5000 in at ratio 2:1. The same pinion in metric clothing (module 25.4/8 = 3.175) would carry a 3.175 mm addendum and, per the printed DIN 867 form, a 3.673 mm dedendum.
Formulas
- ANSI B6.1 full-depth proportions (inch, DP)
D = N/P; DO = (N+2)/P; DR = (N−2.5)/P; a = 1/P; b = 1.25/P; ht = 2.25/P; t = 1.5708/P- Module proportions as printed (DIN 867 table)
D = mN; DO = m(N+2); a = m; b = 1.157m; ht = 2.157m; t = 1.5708m- Mesh geometry (20° PA)
Db = D·cos 20°; C = (N₁+N₂)/(2P) = m(N₁+N₂)/2; ratio = N₂/N₁
Standards & References
- Machinery's Handbook, Spur Gearing — Table 2 "Formulas for Tooth Parts, 20-and 25-degree Involute Full-depth Teeth", ANSI B6.1-1968 (R1974) coarse-pitch tooth forms (all inch proportions verified against the printed table)
- Machinery's Handbook, Module System Gearing — "German Standard Tooth Form for Spur and Bevel Gears DIN 867" table (metric proportions as printed, 0.157m clearance) and the module = 25.4/P equivalence rules
- Machinery's Handbook — printed 18-tooth minimum pinion count for 20° full-depth involute (undercut warning threshold)
Frequently Asked Questions
How do I calculate spur gear outside diameter?
Add 2 to the tooth count and divide by the diametral pitch — OD = (N+2)/P — or multiply by the module: OD = m(N+2). A 24-tooth 8-DP gear blank turns to 3.250 in; a 20-tooth module-2 blank to 44 mm. It is the pitch diameter plus two addendums, and it is the dimension you machine the blank to before cutting teeth.
What is the difference between module and diametral pitch?
They are the same idea upside down: DP counts teeth per inch of pitch diameter, module measures millimetres of pitch diameter per tooth. Converting is exact — module = 25.4 ÷ DP (10 DP ↔ 2.54 module, the Handbook's example) — but hardware is standardized on preferred values in each system, so a converted number usually lands between stock cutters. Gears mesh only with the same pitch/module and pressure angle.
How do I find the center distance between two spur gears?
Half the sum of the pitch diameters: C = (N₁+N₂)/(2P) in the inch system or m(N₁+N₂)/2 metric. A 24- and 48-tooth pair at 8 DP sits at (24+48)/16 = 4.500 in. That is the standard, unshifted distance — designs using profile shift or center-distance adjustment for backlash deviate deliberately.
Why do the inch and metric dedendums differ (1.25/P vs 1.157m)?
Because the printed standards differ. ANSI B6.1's preferred full-depth dedendum is 1.25/P (clearance 0.25/P), while the DIN 867 table Machinery's Handbook prints uses a 0.157m clearance — the value American module-cutter makers standardized — giving b = 1.157m, with a 1.167m variant at one-sixth-module clearance. Many modern ISO-based metric designs cut deeper roots; the cutter or the drawing's standard governs, and this calculator reports each system exactly as its cited table prints it.
What is gear undercut and when does it happen?
When a generating cutter digs into the root flank of a small pinion, thinning the tooth right where bending stress peaks and shortening the usable involute. The Handbook's printed minimum for 20° full-depth teeth is 18 pinion teeth; below that the calculator flags the risk. Real designs fix it with profile shift (enlarged pinion), a modification beyond standard proportions — see a gear design reference. Ratios and RPM/torque live in the gear ratio calculator.