Gear Tooth Calculator: Module, Pitch and Gear Size

A gear tooth calculator works out basic gear dimensions from tooth count, module, and pressure angle. Enter the number of teeth and a gear-size measure, and it can return pitch diameter, outside diameter, circular pitch, and related geometry; the results describe standard spur-gear geometry, not a complete design or strength rating.

Formula

For a standard, unshifted external spur gear:

• Pitch diameter, d = mz: d is pitch diameter in millimetres, m is module in millimetres, and z is the number of teeth.
• Outside diameter, da = m(z + 2): da is the diameter across the tooth tips. This assumes standard addendum equal to one module.
• Circular pitch, p = πm: p is the distance along the pitch circle between corresponding points on adjacent teeth.
• Base diameter, db = d cos α: db is the base-circle diameter and α is the pressure angle, entered in degrees.

The calculator reports diameters and spacing in millimetres. It does not determine tooth width, root diameter, backlash, center distance, or load capacity.

Worked Example

Using the calculator defaults—20 teeth, a 2 mm module, and a 20° pressure angle:

1. Pitch diameter: d = 2 × 20 = 40.00 mm.
2. Outside diameter: da = 2 × (20 + 2) = 44.00 mm.
3. Circular pitch: p = π × 2 = 6.283… mm, which rounds to 6.28 mm.
4. Base diameter: db = 40 × cos(20°) = 37.587… mm, which rounds to 37.59 mm.

These are nominal geometric values under the stated standard-gear assumptions. Manufacturing tolerances and tooth modifications can make a physical gear differ.

What the Measurements Mean

Module is the metric measure of tooth size: it is the pitch diameter divided by the tooth count. A larger module means larger teeth for a given number of teeth. For example, a 20-tooth gear with module 2 has a 40 mm pitch diameter; at module 3, the same tooth count gives 60 mm.

The pitch circle is an imaginary reference circle used to describe how two gears mesh. For a pair of standard external spur gears, their nominal center distance is half the sum of their pitch diameters. If both gears share module m and have tooth counts z1 and z2, then center distance a = m(z1 + z2)/2. A 20-tooth and 40-tooth pair at module 2 therefore has a nominal center distance of 60 mm.

Outside diameter is measured over the tooth tips, so it is not the same as pitch diameter. Circular pitch is a linear distance along the pitch circle. The base circle is important to involute tooth geometry; its diameter depends on both pitch diameter and pressure angle.

How to Choose Inputs and Units

Use the gear’s specified module and tooth count when those are known. If you are measuring a part, count every tooth and measure across the tooth tips with suitable measuring tools, but do not assume that outside diameter alone uniquely identifies module: tooth count, profile, and modifications matter too.

• Module: Use millimetres per tooth as the calculator input. Module is not the same as diametral pitch, which is commonly used in inch-based gear specifications.
• Pressure angle: Enter the angle from the gear specification. Common nominal values include 20° and 14.5°, but a familiar value should not be treated as proof of the actual gear specification.
• Units: Keep dimensions consistent. To convert a result from millimetres to inches, divide by 25.4; to convert inches to millimetres, multiply by 25.4.

For an inch-based gear, diametral pitch P is the number of teeth per inch of pitch diameter, so d = z/P in inches. Module and diametral pitch are related by m = 25.4/P, with m in millimetres. They are alternative sizing conventions, not interchangeable numbers.

When This Calculator Is Useful—and Its Limits

Use these calculations for a first geometry check, comparing catalog specifications, estimating nominal gear size, or checking whether a proposed pair has a plausible center distance. They are most directly applicable to standard external spur gears with ordinary full-depth teeth.

They are not a substitute for a detailed gear design. Helical, bevel, worm, internal, and nonstandard gears have additional geometry or different conventions. Profile shift, stub teeth, custom addendum or dedendum, and manufacturing corrections can change dimensions. A tooth-count and module calculation also says nothing by itself about material, lubrication, noise, wear, speed, or the torque a gear can safely transmit.

For a meshing pair, both gears generally need compatible tooth systems, including matching module and pressure angle. Tooth counts and center distance must also suit the intended arrangement. A calculated diameter does not confirm that two real gears will mesh correctly.

Common Mistakes

• Confusing module with diametral pitch: They use different units and conventions. Convert before using a formula rather than entering one as the other.
• Calling outside diameter the pitch diameter: For a standard external spur gear, outside diameter is typically two modules larger than pitch diameter.
• Mixing units: Do not combine a module in millimetres with a diameter in inches without conversion.
• Rounding too early: Keep full precision in intermediate calculations, then round displayed dimensions to the precision appropriate for the task.
• Assuming all gears are standard: A modified tooth profile can invalidate the standard outside-diameter formula even when tooth count and module are correct.

Frequently Asked Questions

How do I calculate gear pitch diameter?
For a metric gear, multiply module by tooth count: d = mz. With module 2 mm and 20 teeth, pitch diameter is 40 mm.

How do I find a gear’s module?
If pitch diameter and tooth count are known, divide pitch diameter in millimetres by tooth count: m = d/z. Measuring only the outside diameter is less reliable unless the gear’s tooth form and standard dimensions are known.

What is the difference between module and diametral pitch?
Module measures pitch diameter in millimetres per tooth; diametral pitch measures teeth per inch of pitch diameter. Their conversion is m = 25.4/P.

Can two gears mesh if they have the same module?
Matching module is necessary for ordinary metric gears, but it is not enough. Pressure angle, tooth form, and other design details must also be compatible.

Does the calculator tell me whether a gear is strong enough?
No. It calculates basic geometry only. Strength depends on factors such as material, tooth dimensions, load, speed, support, and operating conditions, which require a separate engineering assessment.

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in teeth | Number of teeth | - | 20 | 1
in module | Module | mm | 2 | 0
in angle | Pressure angle | deg | 20 | 0
out pitch_diameter | Pitch diameter | mm | teeth * module | 2
out outside_diameter | Outside diameter | mm | module * (teeth + 2) | 2
out circular_pitch | Circular pitch | mm | pi * module | 2
out base_diameter | Base diameter | mm | pitch_diameter * cosd(angle) | 2