The gear module is calculated by dividing the pitch circle diameter by the number of teeth: m = d ÷ z. For example, the module of a gear with a pitch circle diameter of 120 mm and 40 teeth is 120 ÷ 40 = 3. The module is expressed in millimeters and represents the size of the tooth with a single value; two gears that will mesh together must have the same module.
If you do not have a technical drawing, you cannot directly measure the pitch circle diameter because this circle is not a visible surface on the gear. In this case, measure the outside diameter with a caliper and use the formula m = outside diameter ÷ (z + 2). Below, you will find the formulas, standard module values, the steps for determining the module by measurement, and the differences in helical gears.
Module Formula and Basic Gear Dimensions
Once the module is known, all basic dimensions of a spur gear can be derived from the number of teeth. The following formulas apply to standard gears without profile shift.
- Pitch circle diameter: d = m × z
- Outside diameter: da = m × (z + 2)
- Root diameter: df = m × (z − 2,5)
- Pitch: p = π × m
- Tooth height: h = 2,25 × m
The pitch circle is the theoretical circle on which two gears are assumed to roll against each other without slipping. The outside diameter is the outermost dimension of the gear; the teeth extend outward from the pitch circle by 1 module. The root diameter is measured at the bottom of the tooth space and lies 1.25 modules inside the pitch circle. The 0.25-module difference provides clearance so that the tip of the mating gear tooth does not rub against the root.
Example Calculation
For a spur gear with a module of 3 and 40 teeth:
- Pitch circle diameter: 3 × 40 = 120 mm
- Outside diameter: 3 × (40 + 2) = 126 mm
- Root diameter: 3 × (40 − 2,5) = 112,5 mm
- Pitch: 3,1416 × 3 = 9,42 mm
- Tooth height: 2,25 × 3 = 6,75 mm
If this gear meshes with a 20-tooth pinion, the pinion pitch circle diameter will be 3 × 20 = 60 mm. The center distance between the two gears is half the sum of their pitch circle diameters: (120 + 60) ÷ 2 = 90 mm.
Standard Module Values
The module is not selected randomly. Since gear-cutting tools such as hobs and milling cutters are manufactured according to specific modules, one of the standard values is used in design. For cylindrical gears used in general machinery manufacturing, the ISO 54 standard defines two series; the first series is the preferred choice, while the second series is used when the first series is not suitable.
| Series | Module values (mm) |
|---|---|
| Series 1 (preferred) | 1 – 1,25 – 1,5 – 2 – 2,5 – 3 – 4 – 5 – 6 – 8 – 10 – 12 – 16 – 20 – 25 – 32 – 40 – 50 |
| Series 2 | 1,125 – 1,375 – 1,75 – 2,25 – 2,75 – 3,5 – 4,5 – 5,5 – 7 – 9 – 11 – 14 – 18 – 22 – 28 – 36 – 45 |
The dimensions of a tooth for commonly used modules are shown below. These values are independent of the number of teeth and are the same for all gears with the same module.
| Module | Pitch (mm) | Tooth height (mm) | Outside diameter for 20 teeth (mm) |
|---|---|---|---|
| 1 | 3,14 | 2,25 | 22 |
| 1,5 | 4,71 | 3,375 | 33 |
| 2 | 6,28 | 4,5 | 44 |
| 3 | 9,42 | 6,75 | 66 |
| 4 | 12,57 | 9 | 88 |
| 5 | 15,71 | 11,25 | 110 |
How to Determine the Module of an Existing Gear by Measurement
When replacing a broken or worn gear, you often have only the physical part itself. For a spur gear, a caliper is sufficient to determine the module.
- Count the teeth. Mark the tooth where you started with a pen while counting.
- Measure the outside diameter of the gear with a caliper. Measure at several different points and use the largest value; worn teeth can reduce the measured diameter.
- Divide the measured diameter by the number of teeth plus two: m = da ÷ (z + 2).
- Round the result to the nearest standard module.
Example: You measure the outside diameter of a 25-tooth gear as 67,4 mm. 67,4 ÷ 27 = 2,496; therefore, the gear module is 2,5.
Points to Consider During Measurement
- Gears with an odd number of teeth: If the number of teeth is odd, a tooth space will be directly opposite a tooth and the caliper may read slightly smaller than the actual diameter. In this case, it is more reliable to mount the gear on a shaft and measure the radius from the shaft center to the tooth tip, or calculate it from the center distance between two gears.
- Verification using center distance: If you know the tooth counts of two mating gears and the distance between their centers, you can verify the result using the formula m = 2 × center distance ÷ (z1 + z2).
- If the result does not approach any standard value: The gear may use an imperial measurement system (DP, diametral pitch). The relationship between module and DP is m = 25,4 ÷ DP; for example, a DP 8 gear corresponds to a module of 3,175. Other possibilities include profile shift or a helical gear design.
- Module gauge: In the workshop, you can also quickly check the module by fitting module gauges into the tooth spaces.
Difference Between Normal Module and Transverse Module in Helical Gears
A spur gear has a single module. In a helical gear, because the teeth are angled relative to the axis, the module is defined in two different planes:
- Normal module (mn): The module measured in a section perpendicular to the tooth. This is the module of the gear-cutting tool, and standard values apply to the normal module.
- Transverse module (mt): The module measured on the gear face, in the plane perpendicular to the axis of rotation. This value is used in diameter calculations.
The relationship between them is determined by the helix angle (β): mt = mn ÷ cos β. As the helix angle increases, the transverse module also increases; therefore, a helical gear cut with the same tool will have a larger diameter than a spur gear with the same number of teeth.
- Pitch circle diameter: d = mn × z ÷ cos β
- Outside diameter: da = d + 2 × mn
- Root diameter: df = d − 2,5 × mn
Example: For a gear with a normal module of 2, 30 teeth and a helix angle of 15°, cos 15° = 0,9659. The transverse module is 2 ÷ 0,9659 = 2,071 mm, the pitch circle diameter is 2,071 × 30 = 62,12 mm, the outside diameter is 62,12 + 4 = 66,12 mm and the root diameter is 62,12 − 5 = 57,12 mm. If the same gear were a spur gear, its outside diameter would be 64 mm.
Therefore, if you measure the outside diameter of a helical gear and apply the spur gear formula, you will obtain a non-standard and meaningless module value. For an accurate result, you must also know the helix angle. The structure of helical gears and the applications in which they are preferred are explained in our article what is a helical gear reducer.
Gear Module Calculation Tool
Enter the number of teeth. If you know the module, fill in the module field and the tool will calculate the gear diameters. If you do not know the module, leave that field blank and enter the outside diameter measured with a caliper; the tool will calculate the module. For helical gears, also enter the helix angle; leave it as 0 for spur gears.
The results are valid for standard gears without profile shift. Verify the dimensions against the technical drawing before manufacturing.
What Should Be Considered When Selecting a Module?
When designing a new gear, the module is determined according to the load that the tooth must carry. As the module increases, the tooth becomes thicker and can carry a greater load; however, for the same number of teeth, the gear diameter and weight also increase. A smaller module provides a more compact and quieter structure, but the tooth root is thinner. The final selection is determined through a strength calculation that considers the transmitted load, material, heat treatment and face width together. You can find information about materials on our heat treatment and materials page.
The module only defines the size of the gear tooth; it does not indicate how much a gear pair will reduce speed or how much output torque will be obtained. For these calculations, see our article on gearbox speed calculation and our Gearbox Calculation page.
Frequently Asked Questions About Gear Module Calculation
Can two gears with different modules work together?
No. If the modules are different, the tooth pitches are also different; the teeth will not fit into each other's spaces, causing jamming or breakage within a short period. In helical gears, the helix angle must also be compatible in addition to the normal module.
What changes when the number of teeth increases with the same module?
The tooth size remains the same, but the gear diameter increases. A 20-tooth gear with module 2 has a pitch circle diameter of 40 mm, while a 60-tooth gear has a pitch circle diameter of 120 mm; the tooth height is 4,5 mm in both cases.
What is the difference between module and DP (diametral pitch)?
Both describe tooth size. Module is used in the metric system, and the tooth becomes larger as the module increases. DP is used in the imperial system and represents the number of teeth per inch of pitch diameter; the tooth becomes smaller as DP increases. The conversion formula is m = 25,4 ÷ DP.
How is the module of a rack gear determined?
Since a rack has no diameter, the pitch is measured. Measure the distance between identical points on several consecutive teeth, divide it by the number of tooth intervals to determine the pitch, and then divide the pitch by 3,1416. For example, if 10 tooth intervals measure 94,2 mm, the pitch is 9,42 mm and the module is 3. For information on how the mechanism works, see our article on the rack and pinion mechanism.
Why does the module I measured not match a standard value?
The most common reasons are that the gear is helical, uses imperial dimensions (DP), has been manufactured with profile shift, the diameter has been under-measured due to an odd number of teeth, or the tooth tips are worn. If possible, also measure the center distance together with the mating gear to verify the result.
I want to replace a gear inside a gearbox. Is knowing the module enough?
No. In addition to the module and number of teeth, the helix angle and direction, face width, profile shift, material and heat treatment must also be the same. Therefore, the safest approach is to request the spare part from the manufacturer using the type and serial number on the gearbox nameplate. For more information about the process, read our article on gearbox repair and overhaul.
