Spur Gear Calculator
Work out the full geometry of a pair of standard involute spur gears. Enter a metric module or an inch diametral pitch, the tooth counts and the pressure angle to get pitch, outside, root and base diameters, center distance, tooth depth, chordal thickness and contact ratio, with a to-scale drawing of the mesh.
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Common: 0.5, 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10
Common: 4, 6, 8, 10, 12, 16, 20, 24, 32, 48
Same unit as the system (mm or in). Rule of thumb: 8–16 × module
How to use the Spur Gear Calculator
- Choose the system: metric module (mm) or diametral pitch (teeth per inch).
- Enter the module or DP and pick the pressure angle (20° is the modern standard).
- Type the tooth counts of the pinion (smaller gear) and the gear.
- Add optional values: face width and pinion rpm.
- Read the results: center distance first, then diameters, tooth sizes, contact ratio and the drawing. Download the table as CSV if you need it.
Spur gear formulas
The calculator uses standard full-depth involute proportions (ISO 53 basic rack for metric module gears, the AGMA full-depth system for inch diametral pitch gears) with no profile shift. Here m is the module in mm, P the diametral pitch, z or N the number of teeth and α the pressure angle.
| Dimension | Metric (module) | Inch (diametral pitch) |
|---|---|---|
| Pitch diameter | d = m z | D = N ÷ P |
| Addendum | ha = m | a = 1 ÷ P |
| Dedendum | hf = 1.25 m | b = 1.25 ÷ P |
| Outside diameter | da = m (z + 2) | Do = (N + 2) ÷ P |
| Root diameter | df = m (z − 2.5) | DR = (N − 2.5) ÷ P |
| Base diameter | db = d cos α | Db = D cos φ |
| Circular pitch | p = π m | p = π ÷ P |
| Tooth thickness at pitch circle | s = π m ÷ 2 | t = π ÷ (2P) |
| Center distance | a = m (z₁ + z₂) ÷ 2 | C = (N₁ + N₂) ÷ (2P) |
| Whole depth | h = 2.25 m | ht = 2.25 ÷ P |
Some fine-pitch inch standards (20 DP and finer) use a slightly deeper dedendum of 1.2 ÷ P + 0.002 in; check your cutter or supplier if you work with very small gears.
Common module and diametral pitch sizes
| Module (mm) | Equivalent DP | Circular pitch | Whole depth | Nearest common DP |
|---|---|---|---|---|
| 0.5 | 50.80 | 1.571 mm | 1.125 mm | 48 |
| 0.8 | 31.75 | 2.513 mm | 1.800 mm | 32 |
| 1 | 25.40 | 3.142 mm | 2.250 mm | 24 |
| 1.5 | 16.93 | 4.712 mm | 3.375 mm | 16 |
| 2 | 12.70 | 6.283 mm | 4.500 mm | 12 |
| 2.5 | 10.16 | 7.854 mm | 5.625 mm | 10 |
| 3 | 8.47 | 9.425 mm | 6.750 mm | 8 |
| 4 | 6.35 | 12.566 mm | 9.000 mm | 6 |
| 5 | 5.08 | 15.708 mm | 11.250 mm | 5 |
| 6 | 4.23 | 18.850 mm | 13.500 mm | 4 |
The "nearest" DP is only close in size: module and DP gears never mesh correctly with each other.
Design tips
- Mating gears must match in module (or DP) and pressure angle. Tooth count only changes the size and the ratio.
- Avoid tiny pinions. Below about 17 teeth at 20° the cutter removes material at the tooth root (undercut), weakening the tooth and shortening contact.
- Face width is usually 8 to 16 times the module; wider faces need very accurate shaft alignment.
- Hunting tooth ratios (tooth counts with no common factor, such as 19 and 41) spread wear evenly because each tooth meets every tooth of the other gear.
- Backlash is created by thinning the teeth slightly or opening the center distance a little beyond the theoretical value.
Frequently asked questions
How do I calculate the pitch diameter of a spur gear?
Multiply the module by the number of teeth (d = m × z) for metric gears, or divide the teeth by the diametral pitch (d = N ÷ P) for inch gears. A 40-tooth gear with module 2 has a 80 mm pitch diameter; a 48-tooth, 12 DP gear has a 4 in pitch diameter.
What is the outside diameter of a gear?
For standard full-depth teeth the outside (tip) diameter is the pitch diameter plus two addendums: d + 2m in metric or (N + 2) ÷ P in inch units. This is the size of the gear blank before the teeth are cut.
How is the center distance of two gears found?
For standard gears without profile shift the center distance is half the sum of the pitch diameters: a = m (z₁ + z₂) ÷ 2. A 20-tooth and 40-tooth pair in module 2 runs at 60 mm centers. Spreading the centers slightly adds backlash.
How do I convert between module and diametral pitch?
Module = 25.4 ÷ DP, and DP = 25.4 ÷ module. A 12 DP gear is about module 2.117, so it is not interchangeable with module 2 gears even though they look similar. Gears must have the same module (or DP) and pressure angle to mesh.
What tooth depth does a standard spur gear have?
Full-depth teeth have an addendum of 1 module and a dedendum of 1.25 module, so the whole depth is 2.25 × m (or 2.25 ÷ P in inches). The working depth is 2m and the root clearance 0.25m.
What is the minimum number of teeth to avoid undercut?
In theory 2 ÷ sin²α: about 17 teeth at 20° and 32 teeth at 14.5°. In practice about 14 teeth (20°) still runs well. Fewer teeth need a positive profile shift or a larger pressure angle.
What is a good contact ratio?
The contact ratio is the average number of tooth pairs in mesh. It must be above 1 for continuous motion; 1.2 is a common minimum and 1.4 to 1.8 is typical for standard 20° spur gears with a reasonable number of teeth.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.