Tooling and hardware
Spur gear calculator
Pitch diameter = teeth / DP and outside diameter = (teeth + 2) / DP; a 36-tooth, 24 DP gear is 1.5000 in pitch and 1.5833 in outside diameter.
Technical review pending. Formulas and values are cited. This notice comes down after a machinist review.
Pitch diameter
in
PD = N / DP = 36 / 24 = 1.5000 in
Outside diameter (blank size)
in
OD = (N + 2) / DP = (36 + 2) / 24 = 1.5833 in
Whole depth 0.0937 in by formula (fine pitch, 20 DP and finer); cut to the depth stamped on the cutter. Older cutters may be stamped 2.157 / DP (0.0899 in). Involute cutter #3.
| Dimension | Value | Formula with your numbers |
|---|---|---|
| Pitch diameter | 1.5000 in | PD = N / DP = 36 / 24 = 1.5000 in |
| Outside diameter (blank size) | 1.5833 in | OD = (N + 2) / DP = (36 + 2) / 24 = 1.5833 in |
| Addendum | 0.0417 in | a = 1 / DP = 1 / 24 = 0.0417 in |
| Dedendum | 0.0520 in | b = 1.2 / DP + 0.002 = 1.2 / 24 + 0.002 = 0.0520 in |
| Whole depth, by formula (fine pitch, 20 DP and finer) | 0.0937 in | WD = 2.2 / DP + 0.002 = 2.2 / 24 + 0.002 = 0.0937 in |
| Circular pitch | 0.1309 in | p = pi / DP = 3.1416 / 24 = 0.1309 in |
| Tooth thickness at the pitch line | 0.0654 in | t = pi / (2 x DP) = 3.1416 / (2 x 24) = 0.0654 in |
| Base circle diameter | 1.4095 in | BC = PD x cos(PA) = 1.5000 x cos(20°) = 1.4095 in |
| Center distance | 1.1250 in | C = (N1 + N2) / (2 x DP) = (36 + 18) / (2 x 24) = 1.1250 in |
| Involute cutter (DP, Brown & Sharpe series) | #3 | 35 to 54 teeth |
| Module equivalent | 1.0583 mm | m = 25.4 / DP = 25.4 / 24 = 1.0583 mm |
Source: Machinery's Handbook (spur gearing; 20 degree full-depth involute system); ANSI/AGMA 201.02 coarse-pitch tooth proportions; Boston Gear catalog P-1930-BG, engineering information, spur gears (14.5 degree full depth and fine-pitch whole depth); Martin Sprocket & Gear catalog B1015-4 (14.5 degree full depth and 20 degree stub proportions); Hamrock, Fundamentals of Machine Elements (AGMA fine-pitch dedendum).
How it works
Diametral pitch (inch)
PD = N / DP; OD = (N + 2) / DP; WD = 2.25 / DP (20°), 2.157 / DP (14.5°), 2.2 / DP + 0.002 (20 DP and finer); C = (N1 + N2) / (2 x DP)
| Symbol | Meaning | Unit |
|---|---|---|
| N | Number of teeth | teeth |
| DP | Diametral pitch: teeth per inch of pitch diameter | 1/in |
| PD | Pitch diameter | in |
| OD | Outside diameter, the blank size | in |
| a | Addendum = 1 / DP | in |
| b | Dedendum = 1.25 / DP (20°), 1.157 / DP (14.5°), 1.2 / DP + 0.002 (20 DP and finer) | in |
| WD | Whole depth, by tooth system; cut to the depth stamped on the cutter | in |
| 0.002 | Added clearance on fine-pitch teeth | in |
| p | Circular pitch = pi / DP | in |
| t | Tooth thickness at the pitch line = pi / (2 x DP) | in |
| BC | Base circle diameter = PD x cos(PA) | in |
| PA | Pressure angle | deg |
| N1, N2 | Teeth on the gear and on its mate | teeth |
| C | Center distance | in |
Module (metric)
PD = m x N; OD = m x (N + 2); WD = 2.25 x m; C = m x (N1 + N2) / 2
| Symbol | Meaning | Unit |
|---|---|---|
| m | Module: mm of pitch diameter per tooth | mm |
| a, b | Addendum = m; dedendum = 1.25 x m | mm |
| p, t | Circular pitch = pi x m; tooth thickness = pi x m / 2 | mm |
| DP to m | Conversion between the two systems: m = 25.4 / DP | mm |
| System | Addendum a | Dedendum b | Whole depth WD |
|---|---|---|---|
| 20° full depth, coarser than 20 DP (AGMA) | 1 / DP | 1.25 / DP | 2.25 / DP |
| 14.5° full depth, coarser than 20 DP | 1 / DP | 1.157 / DP | 2.157 / DP |
| Fine pitch, 20 DP and finer | 1 / DP | 1.2 / DP + 0.002 in | 2.2 / DP + 0.002 in |
| 20° stub (one catalog source; not calculated here) | 0.8 / DP | 1 / DP | 1.8 / DP |
| Module, 20° | m | 1.25 x m | 2.25 x m |
DP in teeth per inch, m in mm. The calculator picks the row from the pressure angle and the pitch. For 20° coarse pitch, AGMA uses a 1.25 / DP dedendum (2.25 / DP whole depth); some gear catalogs list 2.157 / DP whole depth for 20° as well.
| Cutter | Cuts gears with |
|---|---|
| #1 | 135 teeth to rack |
| #2 | 55 to 134 teeth |
| #3 | 35 to 54 teeth |
| #4 | 26 to 34 teeth |
| #5 | 21 to 25 teeth |
| #6 | 17 to 20 teeth |
| #7 | 14 to 16 teeth |
| #8 | 12 to 13 teeth |
One cutter covers a range of tooth counts when cutting on a dividing head. Pick the cutter by the teeth on the gear being cut.
Worked example: 24 DP, 36 teeth, mating 18 teeth
DP = 24, N = 36, mating gear 18 teeth, 20° full depth. 24 DP is fine pitch (20 DP and finer).
- PD = 36 / 24 = 1.5000 in.
- OD = 38 / 24 = 1.5833 in.
- Whole depth = 2.2 / 24 + 0.002 = 0.0917 + 0.002 = 0.0937 in.
- Circular pitch = 3.1416 / 24 = 0.1309 in; tooth thickness = 0.1309 / 2 = 0.0654 in.
- C = (36 + 18) / (2 x 24) = 54 / 48 = 1.1250 in.
- 36 teeth falls in the 35 to 54 range: cutter #3.
Result: turn the blank to 1.5833 in and cut to the depth stamped on the #3 cutter (2.2 / 24 + 0.002 = 0.0937 in by the formula); centers 1.1250 in apart.
Worked example: module 2, 30 teeth, mating 45 teeth
m = 2 mm, N = 30, mating gear 45 teeth, 20° full depth.
- PD = 2 x 30 = 60.00 mm.
- OD = 2 x 32 = 64.00 mm.
- Whole depth = 2.25 x 2 = 4.50 mm.
- Circular pitch = 3.1416 x 2 = 6.28 mm.
- C = 2 x (30 + 45) / 2 = 2 x 75 / 2 = 75.00 mm.
- Base circle = 60 x cos(20°) = 60 x 0.93969 = 56.38 mm.
Result: 64.00 mm blank, 4.50 mm whole depth, centers 75.00 mm apart.
Worked example: 10 DP, 30 teeth, 14.5° full depth, mating 20 teeth
DP = 10, N = 30, mating gear 20 teeth, 14.5° full depth (coarser than 20 DP).
- PD = 30 / 10 = 3.0000 in; OD = 32 / 10 = 3.2000 in.
- Dedendum = 1.157 / 10 = 0.1157 in; whole depth = 2.157 / 10 = 0.2157 in.
- Base circle = 3.0000 x cos(14.5°) = 3.0000 x 0.96815 = 2.9044 in.
- C = (30 + 20) / (2 x 10) = 50 / 20 = 2.5000 in.
- 30 teeth falls in the 26 to 34 range: cutter #4, in a 14.5° cutter.
Result: 3.2000 in blank, 0.2157 in whole depth, centers 2.5000 in apart.
Shop notes
- Turn the blank to the outside diameter and cut to whole depth from the blank surface.
- At 20 DP and finer the calculator uses the fine-pitch whole depth, 2.2 / DP + 0.002 in (0.0937 in at 24 DP). Older form cutters are stamped 2.157 / DP (0.0899 in at 24 DP). Cut to the depth stamped on the cutter.
- For 20° coarse pitch, AGMA uses a 1.25 / DP dedendum (2.25 / DP whole depth), the value used here; some gear catalogs list 2.157 / DP for 20° as well.
- A gear tooth vernier reads chordal thickness; set it with the chordal addendum from Machinery's Handbook.
- On a 40:1 dividing head, turns per tooth = 40 / N: 36 teeth is 40 / 36 = 1 1/9 turns per tooth.
- 24 DP is about module 1.06 (25.4 / 24). DP and module gears do not mesh with each other.
- Cutter numbers here follow the Brown & Sharpe series for DP cutters. Module cutter sets number their ranges differently, so the calculator gives no number in module mode; check the tooth range stamped on the module cutter.
- Match the cutter's pressure angle to the gear: 14.5° and 20° cutters both carry these numbers.
FAQ
How do you find the pitch diameter of a spur gear?
Divide the number of teeth by the diametral pitch: PD = N / DP. 36 teeth at 24 DP is 1.5000 in.
What blank diameter for a 36-tooth, 24 DP gear?
The outside diameter: (36 + 2) / 24 = 1.5833 in.
How deep do you cut a 24 DP gear?
To the depth stamped on the cutter. 24 DP is fine pitch (20 DP and finer), so the formula gives 2.2 / 24 + 0.002 = 0.0937 in. Older cutters stamped 2.157 / DP give 0.0899 in, so the stamp governs.
What is the whole depth of a 14.5° gear?
2.157 / DP for 14.5° full depth coarser than 20 DP, with a 1.157 / DP dedendum: 0.2157 in at 10 DP. At 20 DP and finer the fine-pitch depth, 2.2 / DP + 0.002 in, applies.
Which involute cutter for 36 teeth?
#3, which covers 35 to 54 teeth.
How do you convert DP to module?
m = 25.4 / DP. 24 DP is about module 1.06 (1.0583 mm).
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