What is a gear module?

In the metric system, the gear module (m) is the unit of size that indicates how big or small a gear tooth is. It is defined as the pitch diameter divided by the number of teeth (m = d / N). Gears must have the same module to mesh together properly.

How do you calculate the outside diameter of a gear?

For a standard spur gear, the outside diameter (also called addendum diameter) is calculated as: da = m × (N + 2), where m is the module and N is the number of teeth.

What is the difference between Module and Diametral Pitch?

Module is the ISO metric standard (mm of pitch diameter per tooth). Diametral Pitch (DP) is the Imperial standard (number of teeth per inch of pitch diameter). They are inversely related: Module = 25.4 / DP.

Mechanical Engineering Calculator

Gear Module & Dimensions Calculator

Calculate comprehensive manufacturing dimensions for ISO metric spur gears, including pitch diameter, outside diameter, root diameter, and circular pitch based on Module (m) and Teeth (N).

Standard Tooth Profile & Geometric Terminology

Visual representation of a standard ISO metric spur gear profile showing root, pitch, and addendum boundaries.

Input Parameters

Module (m)

Standard ISO tooth size indicator.

Number of Teeth (N)

teeth

Total number of teeth on the gear.

Machining Tip

When turning the gear blank on a lathe, always use the Outside Diameter (da). The hobbing cutter will automatically form the pitch and root diameters.

Pitch Diameter (d)

60.00mm

Outside Diameter (da)

64.00 mm

Root Diameter (df)

55.00 mm

Circular Pitch

6.283 mm

Addendum

2.00 mm

Dedendum

2.50 mm

Governing ISO Formulas

Pitch Diameter (d)mm

m × N

Outside Diameter (da)mm

m × (N + 2)

Root Diameter (df)mm

m × (N - 2.5)

Circular Pitch (p)mm

π × m

Addendum (a)mm

1.0 × m

Dedendum (b)mm

1.25 × m

Engineering Code

Integrate full gear geometry generation into your scripts.

Python

import math

def calculate_gear_dimensions(m, N):
    """
    Calculate standard ISO metric spur gear dimensions.
    
    m: Module (mm)
    N: Number of Teeth
    
    Returns: Dictionary of gear dimensions in mm
    """
    if m <= 0 or N < 1:
        raise ValueError("Module must be > 0 and Teeth >= 1")

    return {
        "pitch_dia": m * N,
        "outside_dia": m * (N + 2),
        "root_dia": m * (N - 2.5),
        "circular_pitch": math.pi * m,
        "addendum": 1.0 * m,
        "dedendum": 1.25 * m,
        "whole_depth": 2.25 * m
    }

# Example Usage
m = 2.0
N = 30
dims = calculate_gear_dimensions(m, N)

print(f"Pitch Diameter: {dims['pitch_dia']:.2f} mm")
print(f"Outside Diameter: {dims['outside_dia']:.2f} mm")
print(f"Root Diameter: {dims['root_dia']:.2f} mm")

MATLAB

function dims = gear_dimensions(m, N)
    % Calculate ISO metric spur gear dimensions
    % m = Module (mm)
    % N = Number of Teeth

    if m <= 0 || N < 1
        error('Module must be > 0 and Teeth >= 1');
    end

    dims.pitch_dia = m * N;
    dims.outside_dia = m * (N + 2);
    dims.root_dia = m * (N - 2.5);
    dims.circular_pitch = pi * m;
    dims.addendum = 1.0 * m;
    dims.dedendum = 1.25 * m;
    dims.whole_depth = 2.25 * m;
end

% Example
m = 2.0;
N = 30;
dims = gear_dimensions(m, N);

fprintf('Pitch Diameter: %.2f mm\n', dims.pitch_dia);
fprintf('Outside Diameter: %.2f mm\n', dims.outside_dia);
fprintf('Root Diameter: %.2f mm\n', dims.root_dia);

Excel Formulas

Pitch Dia: =m*N
Outside Dia: =m*(N+2)
Root Dia: =m*(N-2.5)
Circular Pitch: =PI()*m

Example Calculation

Let's design a standard metric spur gear with a Module (m) of 2.0 mm and 30 teeth (N). We need to find the Pitch Diameter (d) and the Outside Diameter (da) for the lathe turning operation.

Pitch Diameter (d)

d = 2.0 × 30

60.0 mm

Outside Diameter (da)

da = 2.0 × (30 + 2)

64.0 mm

Understanding Gear Module and Geometry

In the metric system, the Module (m) is the fundamental parameter that defines the size of a gear tooth. Two gears must have the identical module (and identical pressure angle) to mesh together correctly. Unlike the imperial Diametral Pitch (DP) which counts teeth per inch, the Module represents the millimeters of pitch diameter per tooth.

Standard Gear Dimensions

Once the Module ($m$) and Number of Teeth ($N$) are selected, all other physical dimensions of a standard spur gear can be derived using pure geometry (assuming a standard 20° pressure angle):

Pitch Diameter (d = m × N): The theoretical diameter where two gears physically meet and roll against each other without slipping.
Outside / Addendum Diameter (da = m × (N + 2)): The absolute outermost diameter of the gear. This is the dimension a machinist uses to turn the blank cylinder before cutting the teeth.
Root / Dedendum Diameter (df = m × (N - 2.5)): The diameter at the deepest part of the gear tooth valley. The 0.25m difference provides the necessary clearance to prevent the tip of the mating gear from jamming into the root.
Circular Pitch (p = π × m): The arc distance measured along the pitch circle from a point on one tooth to the corresponding point on the adjacent tooth.

ISO Standard Module Series

To minimize tooling costs (hobbing cutters), engineers should always specify standard ISO modules from Series 1 if possible. Common Series 1 modules include: 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, etc. Series 2 (e.g., 1.125, 2.75) should only be used when strictly necessary.

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Real-World Engineering Cases

Incompatible Replacement Gears

A maintenance team attempted to replace a broken imported 3D printer gear. They counted 40 teeth and measured a 42 mm outside diameter. They ordered a '40 Tooth, 32-Pitch (Imperial)' gear, which has a 33.3 mm outside diameter. It completely failed to mesh.

Engineering Lesson

You cannot mix Metric Module and Imperial Diametral Pitch. The broken gear had a Module of 1.0 (da = 1 × (40+2) = 42 mm). Reverse engineering requires understanding the foundational formulas.

Interference in Low Tooth Count Gears

A designer specified a standard 20° pressure angle pinion with only 12 teeth to achieve a massive reduction ratio. When manufactured, the gear teeth were severely 'undercut' at the root, making them weak and prone to breaking under load.

Engineering Lesson

The standard formulas assume a minimum number of teeth (usually 17 for a 20° pressure angle) to avoid undercutting. For fewer teeth, profile shifting (modifying the addendum) is mandatory.

Frequently Asked Questions

What is the difference between Module and Circular Pitch?

The module is a rational number (like 2.0 or 3.0) representing diameter/teeth. Circular pitch is the physical arc length between teeth along the pitch circle. They are related by the formula: Circular Pitch = π × Module.

How do I measure the module of an unknown gear?

Count the number of teeth (N). Use calipers to carefully measure the Outside Diameter (da). The Module (m) is approximately da / (N + 2). Match this result to the nearest standard ISO Module.

What is the Addendum and Dedendum?

The addendum (a = 1 × m) is the radial distance from the pitch circle to the top of the tooth. The dedendum (b = 1.25 × m) is the radial distance from the pitch circle to the bottom of the tooth root.