What is the gear module?

The module (m) is the ratio of pitch diameter to the number of teeth (m = d / N). It defines the size of the teeth and must be the same for two gears to mesh correctly. Common modules include 1, 1.25, 1.5, 2, 2.5, and 3 mm.

What is the standard pressure angle?

The most common pressure angles for spur gears are 20 degrees (modern standard) and 14.5 degrees (older standard). The pressure angle affects tooth strength and contact ratio. 20 degrees provides stronger teeth.

What is addendum and dedendum?

Addendum is the radial distance from the pitch circle to the tooth tip (= 1 × module). Dedendum is the radial distance from the pitch circle to the tooth root (= 1.25 × module for standard gears), providing clearance for meshing gears.

How is whole depth calculated?

Whole depth = addendum + dedendum = 2.25 × module for standard full-depth teeth. It represents the total height of a tooth from tip to root.

What is circular pitch?

Circular pitch is the distance from a point on one tooth to the corresponding point on the adjacent tooth, measured along the pitch circle. It equals pi times the module (p = π × m).

What is the minimum number of teeth to avoid undercutting?

For a standard full-depth gear cut by a rack, the theoretical minimum is 2 ÷ sin²α: 17.1 at 20° (17 teeth is the commonly quoted minimum), 31.9 at 14.5° (32 teeth) and 11.2 at 25°. Fewer teeth need a positive profile shift of at least x = 1 − N sin²α ÷ 2, for example x ≈ 0.30 for a 12-tooth 20° pinion.

How do I find the module or DP of an existing gear?

Count the teeth and measure the outside (tip) diameter. Module = OD ÷ (N + 2) in mm, or DP = (N + 2) ÷ OD in inches. A 24-tooth gear measuring 52 mm across the tips has module 52 ÷ 26 = 2. If the result is not close to a standard value, the gear may be profile shifted, stub tooth or helical.

How do you calculate gear ratio and center distance from teeth?

Gear ratio = driven teeth ÷ driving teeth, so a 20-tooth pinion driving a 50-tooth gear gives 2.5 : 1 (output turns 2.5 times slower with 2.5 times the torque, ignoring losses). Center distance for standard spur gears = module × (N₁ + N₂) ÷ 2 = 2 × 70 ÷ 2 = 70 mm, or (N₁ + N₂) ÷ (2 × DP) in inches.

How do module and diametral pitch convert?

Module (mm) = 25.4 ÷ diametral pitch (teeth per inch). So 12 DP ≈ module 2.117, 10 DP ≈ 2.54 and module 2 ≈ 12.7 DP. Module and DP gears are generally not interchangeable unless the values happen to match closely.

Gear Tooth Calculator

Enter tooth count and module (mm) or diametral pitch (in) to get every spur gear dimension; add the mating gear for ratio and center distance, or measure an existing gear to identify its module or DP.

For a standard spur gear with N teeth and module m: pitch diameter d = N × m, outside diameter = (N + 2) × m, root diameter = (N − 2.5) × m, whole depth = 2.25 × m and circular pitch = π × m.

Looking for bicycle gearing (gear inches, gain ratio, speed at cadence)? Use the bicycle gear calculator.

Gear Inputs

Enter teeth count, module, and pressure angle to calculate all spur gear parameters.

Number of Teeth (N)

Integer value, typically 3–500

Pitch system

Module (m) — mm

Pitch diameter divided by number of teeth

Pressure Angle (°)

Mating gear (optional)

Driven teeth N₂

or target ratio

Input rpm

Calculated Parameters

All dimensions in millimeters unless noted.

Enter values and click Calculate to see results.

Identify a gear: find module or DP from teeth and outside diameter

Count the teeth and measure across the tooth tips with calipers.

Teeth counted

Outside diameter

Unit

Standard modules and DP equivalents

Module (mm)	Equivalent DP	Circular pitch (mm)	Whole depth (mm)
0.5	50.80	1.571	1.125
0.8	31.75	2.513	1.800
1	25.40	3.142	2.250
1.25	20.32	3.927	2.813
1.5	16.93	4.712	3.375
2	12.70	6.283	4.500
2.5	10.16	7.854	5.625
3	8.47	9.425	6.750
4	6.35	12.566	9.000
5	5.08	15.708	11.250
6	4.23	18.850	13.500
8	3.18	25.133	18.000
10	2.54	31.416	22.500

Minimum teeth to avoid undercut

Pressure angle	2 / sin²α	Min teeth
14.5°	31.9	32
20°	17.1	17
25°	11.2	11

Full-depth teeth generated by a rack, no profile shift. Below the minimum, use a positive profile shift x ≥ 1 − N sin²α / 2.

Spur Gear Formula Reference

Pitch Diameter

d = N × m

Addendum

a = 1.000 × m

Dedendum

b = 1.250 × m

Whole Depth

h = 2.250 × m

Outside Diameter

dₐ = (N + 2) × m

Root Diameter

dₑ = (N − 2.5) × m

Circular Pitch

p = π × m

Base Circle Diameter

dᵇ = d × cos(φ)

Tooth Thickness

t = π × m / 2

Diametral Pitch

DP = 25.4 / m (d = N / DP in)

Gear Ratio

i = N₂ / N₁

Center Distance

a = m × (N₁ + N₂) / 2

φ = pressure angle. Formulas follow ISO 21771 / AGMA 913 standards for full-depth spur gear teeth.

Quick answer

For a standard spur gear with N teeth and module m: pitch diameter d = N × m, outside diameter = (N + 2) × m, root diameter = (N − 2.5) × m, whole depth = 2.25 × m and circular pitch = π × m. In inches use diametral pitch: d = N ÷ DP, with m = 25.4 ÷ DP. Two gears mesh only with the same module and pressure angle; ratio = N₂ ÷ N₁ and center distance = m(N₁ + N₂) ÷ 2. At 20° a pinion needs about 17 teeth to avoid undercut.

Examples

20 teeth, module 2, 20°	Value
Pitch diameter	40 mm
Outside diameter	44 mm
Root diameter	35 mm
With a 50-tooth gear: ratio	2.5 : 1
Center distance	70 mm

Enter teeth and module or DP for all dimensions, add the mating gear for ratio and center distance, or measure a gear to find its module.

Summary

How it works

Enter the number of teeth on the gear.
Choose module (mm) or diametral pitch (teeth per inch) and enter its value.
Optionally enter a pressure angle (default is 20 degrees).
Click Calculate to see all spur gear parameters instantly.
Optionally enter the mating gear's teeth (or a target ratio) and input speed for the ratio, output rpm and center distance.
To identify an unknown gear, count its teeth and measure the outside diameter: module = OD ÷ (N + 2), DP = (N + 2) ÷ OD in inches.
Use the Reset button to clear inputs and start over.

Use cases

Verify gear dimensions during mechanical design review.
Quickly prototype gear sets for robotics or machinery projects.
Teach gear geometry to engineering students.
Cross-check CNC machining drawings for spur gears.
Determine outside and root diameters for gear blanks.
Calculate circular pitch for gear mesh compatibility.

Frequently Asked Questions

Last updated: 2026-10-10 · Reviewed by Nham Vu