The module m is the key parameter for any gear system. It describes the size of the individual teeth and thus determines whether two gears can mesh at all: Only gears with the same module mesh with each other.
The basic formula is straightforward: m = d / z — pitch circle diameter divided by the number of teeth, result in millimeters. All other gear dimensions follow from this: circular pitch, tip circle, root circle, and centre distance. Which standard modules DIN 780 provides and what matters when selecting a module is covered in the sections below.
Summary: Key Formulas
- Module: m = d / z
- Circular pitch: p = π · m
- Pitch circle diameter: d = m · z
- Tip circle diameter: dₐ = d + 2 · m
- Root circle diameter: d_f = d − 2.5 · m (standard gearing)
- Center distance (gear pair): a = m · (z₁ + z₂) / 2
Basics: Module, Circular Pitch, and Gear Diameters
The gear geometry of a spur gear can be derived entirely from three basic parameters: module m, number of teeth z, and pressure angle α (standard value: 20°). The module is the only value with the dimension of length — all other dimensions are obtained by multiplying the module by a dimensionless number.
Module m
The module m is defined as the quotient of pitch circle diameter d and the number of teeth z:
The module specifies the tooth size: A module of 2 mm means that each tooth occupies 2 mm of the pitch circle circumference. Larger modules result in larger, more durable teeth—but fewer teeth for a given diameter.
Circular pitch p
The circular pitch p is the distance between two adjacent like flanks measured on the pitch circle:
Two gears mesh properly only if they have the same pitch—and since p = π · m, this means that both gears have the same module.
Circular dimensions of standard gear teeth
All diameters can be calculated based on the module and number of teeth (standard gearing without profile shift, pressure angle 20°):
- Pitch circle d = m · z — the reference circle on which the pitch p lies
- Tip circle dₐ = d + 2 · m = m · (z + 2) — outside diameter of the gear
- Root circle d_f = d − 2.5 · m = m · (z − 2.5) — base of the tooth root
Further reading: Pitch circle · Module
Step-by-step calculation
Two of the three values m, d, and z are always known — the third follows directly. From there, tip circle, root circle, and circular pitch are derived.
Calculation example: Module m = 2, number of teeth z = 20
Given: Module m = 2 mm, number of teeth z = 20. Find: all gear diameters and the circular pitch.
- Pitch circle: d = m · z = 2 · 20 = 40 mm
- Tip circle: dₐ = d + 2 · m = 40 + 4 = 44 mm
- Root circle: d_f = d − 2.5 · m = 40 − 5 = 35 mm
- Circular pitch: p = π · m = π · 2 ≈ 6.28 mm
Center distance of a gear pair
For two meshing gears with the same module m and tooth counts z₁ and z₂, the following applies:
Example: m = 2, z₁ = 20, z₂ = 40 → a = 2 · (20 + 40) / 2 = 60 mm. Further details in the glossary entry Centre Distance.
Overview Table: Circular Dimensions for Various Combinations
| Module m | Number of teeth z | d = m × z [mm] | dₐ [mm] | d_f [mm] |
|---|---|---|---|---|
| 1 | 20 | 20 | 22 | 17.5 |
| 2 | 20 | 40 | 44 | 35 |
| 2 | 40 | 80 | 84 | 75 |
| 3 | 25 | 75 | 81 | 67.5 |
| 4 | 18 | 72 | 80 | 62 |
Highlighted line = calculation example from the text. d_f based on standard gearing (dedendum = 1.25 · m).
The gear ratio of a gear pair is given by the ratio of the number of teeth: i = z₂ / z₁. The gear ratio calculator helps with quick sizing. The remaining gear dimensions — pitch circle diameter, tip circle diameter, and root circle diameter — are provided by the gear geometry calculator.
Determining the module of an unknown gear
If a gear is at hand without a drawing, the module can be back-calculated from the tip diameter and the number of teeth. For a standard external spur gear, dₐ = m · (z + 2), from which it follows:
m ≈ dₐ / (z + 2)
dₐ = tip diameter [mm] | z = counted number of teeth
How to proceed:
- Count the teeth → z
- Measure the tip diameter dₐ with a caliper (for an even number of teeth directly across two opposite tooth tips; for an odd number of teeth via the base tangent (span) measurement or by calculation from the radius)
- Calculate m ≈ dₐ / (z + 2)
- Round to the nearest standard module per DIN 780, series 1
Example: dₐ = 66 mm, z = 20 → m ≈ 66 / 22 = 3 mm
The approximation applies to standard spur gears without profile shift. For helical gears, dₐ yields only the normal module via the helix angle; for profile-shifted gears, dₐ deviates from m · (z + 2) — when in doubt, verify the result via the base tangent (span) measurement.
DIN 780 Standard Series: Preferred Modules
DIN 780 specifies which module values should be used preferentially. This standardization ensures the interchangeability of gears across manufacturers and simplifies inventory management. There are two series: Series 1 is the preferred choice; Series 2 may only be selected if no module from Series 1 fits.
| Series | Standard modules (selection) [mm] | Note |
|---|---|---|
| Series 1 (preferred) | 1 · 1.25 · 1.5 · 2 · 2.5 · 3 · 4 · 5 · 6 · 8 · 10 · 12 · 16 · 20 | Always choose first |
| Series 2 (fallback) | 1.125 · 1.375 · 1.75 · 2.25 · 2.75 · 3.5 · 4.5 · 5.5 · 7 · 9 · 11 · 14 · 18 | Only if Series 1 does not fit |
In practice, modules 1 through 6 from Series 1 cover the majority of mechanical engineering applications. Anyone combining standard gears from different suppliers must ensure that, in addition to the module, the pressure angle (standard 20°) also matches. An overview of gear geometry: Basic Concepts of Gear Technology.
Select a module based on load and installation space
The choice of module is not simply a matter of calculations, but rather a compromise between load-bearing capacity and installation space. A larger module means:
- Advantages: greater tooth root height, higher bending strength, better resistance to pitting, greater resistance to shock loads
- Disadvantages: fewer teeth for a given diameter, resulting in higher noise levels and less smooth running (contact ratio decreases); more installation space required for the same tooth count
Practical guidelines:
- Select the smallest module that satisfies the strength condition.
- Maintain a tooth count of ≥ 17 to prevent undercut (at a 20° pressure angle).
- Prefer modules from the DIN 780 Series 1 for better availability.
- For smooth-running, high-speed gearboxes, use a smaller module with a high tooth count (high contact ratio).
For a complete design of a spur gear transmission—including gear ratio, center distance, and material selection—we recommend consulting the guide Spur Gearboxes: Fundamentals & Design.
Common mistakes when selecting modules
Error 1: Module mismatch – gears do not mesh
The classic assembly and spare parts mistake: Two gears are combined that look similar but have different modules. Since the pitch p = π · m is different for each module, the teeth do not mesh—the gears either jam or run with severe wear. Always verify the module and pressure angle of both gears.
Error 2: Undercut caused by too few teeth
With a pressure angle of 20° and standard gearing, undercut (a notch at the tooth root) occurs when the tooth count falls below approximately 17. Undercutting weakens the tooth root and reduces the contact ratio. Solution: Increase the number of teeth, reduce the module, or apply a profile shift. Learn more in the guide Basic Concepts of Gear Technology.
Error 3: Non-standard module selected
Choosing a module outside the DIN 780 series limits interchangeability and makes it much more difficult to source replacement parts. Standard modules from Series 1 are almost always in stock — custom modules are not. Deviating from the standard is only worthwhile for very specific space requirements. Need a gear with a non-standard module? TEA supplies gears to drawing.
Error 4: Module selected without a strength analysis
The module alone does not indicate the load-carrying capacity of a specific gear—material, heat treatment, tooth width, and operating loads determine its service life. Selecting a module that is too small can lead to root fracture despite geometrically correct meshing. Design in accordance with DIN 3990 or ISO 6336 is essential for power-transmitting gearboxes. Learn more in the article Selecting a gear: Material, module, and load capacity.
Practical Tip from TEA:
In our consulting work, the most common issue we see with spare-part enquiries is that the module is determined via the formula m ≈ dₐ / (z + 2) and the result is then rounded—without verification—to an odd value such as 2.1 or 3.2. Such values do not exist in DIN 780; almost always there is a worn tip circle behind it, or helical gearing, in which dₐ only yields the normal module via the helix angle β. Rule of thumb: if the calculated result is close to a Series 1 module, that is exactly the one intended; if it deviates significantly, before submitting the enquiry also measure the base tangent (span) across several teeth and—if available—state the helix angle, otherwise the replacement gear will not mesh.
Custom special-purpose gears?
TEA manufactures and supplies gears, pinions, and custom-toothed components in all standard modules—including custom-made parts with individually calculated modules.
View custom gear solutions →Related articles
Selecting a gear: Material, module, and load capacity
From load requirements to the right gear: a concise guide to material selection, module sizing, and load capacity verification.
Helical gearing vs. spur gearing
Smooth operation, axial force, and load capacity: When is helical gearing preferable to spur gearing?
Gear Technology: Topic Overview
An overview of all guides, fundamentals, and tools related to gears and gear technology.