Gear Diameter Calculator: Pitch & Outer Size

This gear diameter calculator works out the pitch diameter and outer (tip) diameter of a spur gear from the number of teeth and the module (metric) or diametral pitch (imperial). You enter tooth count and module (or DP), and you get the pitch diameter, outer diameter, and root diameter in millimeters or inches.

Formula

For metric gears (module system), the pitch diameter is simply the number of teeth multiplied by the module:

Pitch diameter (d) = N × m

• N = number of teeth (dimensionless)
• m = module in millimeters (pitch per tooth)

The outer diameter (tip diameter) adds two modules because the addendum (height of the tooth above the pitch circle) equals one module:

Outer diameter (da) = (N + 2) × m

The root diameter subtracts a clearance allowance. Standard full-depth gears use a dedendum of 1.25 × module, so:

Root diameter (df) = (N − 2.5) × m

For imperial gears, the diametral pitch (P) is the number of teeth per inch of pitch diameter. Then:

Pitch diameter (in) = N / P
Outer diameter (in) = (N + 2) / P

All formulas assume standard full-depth involute teeth (pressure angle 20° or 14.5°). They do not apply to helical, bevel, or worm gears.

Worked Example

Using the defaults above: N = 20 teeth, module m = 2 mm.

1. Pitch diameter = 20 × 2 = 40.00 mm
2. Outer diameter = (20 + 2) × 2 = 22 × 2 = 44.00 mm
3. Root diameter = (20 − 2.5) × 2 = 17.5 × 2 = 35.00 mm

The calculator shows exactly these values: pitch 40.00 mm, outer 44.00 mm, root 35.00 mm.

When to Use Each Diameter

• Pitch diameter – used for gear ratios, center distance between two meshing gears, and speed calculations. It is the theoretical rolling circle.
• Outer diameter – needed for blank size when machining a gear, for clearance with the housing, and for checking that the gear fits in the available space.
• Root diameter – important for strength calculations and for ensuring the tooth fillet does not interfere with the mating gear’s tip.

For a pair of gears, the center distance equals half the sum of their pitch diameters. For example, a 20-tooth gear (pitch Ø 40 mm) meshing with a 40-tooth gear (pitch Ø 80 mm) needs a center distance of (40 + 80)/2 = 60 mm.

Common Mistakes and Misconceptions

Mixing module and diametral pitch. Module is in millimeters; diametral pitch is in inches�¹. A gear with module 2 is NOT the same as one with DP 2. To convert: module = 25.4 / DP. So DP 10 equals module 2.54 mm.

Using the outer diameter instead of pitch diameter for ratios. The gear ratio is based on pitch diameters (or tooth counts), not outer diameters. Using outer diameter will give a wrong ratio, especially with small tooth counts.

Forgetting the addendum for the outer diameter. Many people think outer diameter = pitch diameter + one tooth height, but the addendum is one full module, so you add two modules total (one on each side).

Assuming all gears have standard addendum. Some gears are “profile shifted� (modified) to avoid undercutting or to adjust center distance. For those, the outer diameter is different: da = (N + 2 + 2x) × m, where x is the profile shift coefficient. This calculator assumes x = 0.

Limits of the Formula

The formulas above apply to standard external spur gears with full-depth teeth. They do not apply to:

• Internal gears (teeth on the inside of a ring)
• Helical gears (need helix angle correction)
• Bevel or worm gears
• Gears with profile shift (non-standard addendum)

For helical gears, the pitch diameter is N × m / cos(β), where β is the helix angle. For internal gears, the outer diameter of the internal gear is (N − 2) × m (subtract instead of add).

Also, the root diameter formula assumes a standard dedendum of 1.25 × m. Some standards use 1.20 × m or 1.40 × m for fine pitch gears. Always check the gear standard you are using (ISO, AGMA, DIN, etc.).

How to Choose the Right Module or DP

If you are designing a gear pair, you typically know the center distance and the desired gear ratio. The module is chosen based on strength and manufacturing constraints:

1. Determine the required center distance (a) and ratio (i = N2/N1).
2. Pitch diameters: d1 = 2a/(i+1), d2 = 2a·i/(i+1).
3. Choose a standard module (e.g., 1, 1.25, 1.5, 2, 2.5, 3, 4, 5 mm) that gives integer tooth counts: N1 = d1/m, N2 = d2/m.
4. Check for undercutting: for 20° pressure angle, the minimum teeth to avoid undercut is about 17. If N1 is less, consider profile shift or a smaller module.

Larger modules give stronger teeth but require bigger gears for the same tooth count. Smaller modules give smoother, quieter operation but are weaker.

Frequently Asked Questions

What is the difference between pitch diameter and outer diameter? The pitch diameter is the theoretical circle where two meshing gears make contact; it determines the gear ratio. The outer diameter is the physical outside edge of the gear, which includes the tooth height (addendum). The outer diameter is always larger than the pitch diameter by two addendum heights (2 × module).

How do I convert module to diametral pitch? Module (mm) = 25.4 / DP. For example, DP 8 = 25.4/8 = 3.175 mm module. Conversely, DP = 25.4 / module. Use the same conversion for pitch diameters: 1 inch = 25.4 mm.

Can I use this calculator for helical gears? No, not directly. Helical gears have a helix angle that increases the pitch diameter. The formula becomes d = N × m / cos(β), where β is the helix angle. The outer diameter also depends on the helix angle. You need a separate calculator for helical gears.

What is the minimum number of teeth to avoid undercutting? For standard 20° pressure angle gears, the minimum is 17 teeth. For 14.5° pressure angle, it is about 32 teeth. Below these numbers, the tooth root becomes weakened and the gear may have interference. You can use profile shift to go lower.

Why does my gear have a different outer diameter than the formula gives? The most common reasons are: the gear has profile shift (modified addendum), it is a non-standard tooth form, or it is not a full-depth gear. Some gears use stub teeth (shorter than full-depth) or long addendum. Check the gear drawing or the manufacturer’s spec.

Does the pressure angle affect the diameter? No, the pitch diameter and outer diameter depend only on tooth count and module (or DP). The pressure angle affects the tooth shape, strength, and minimum tooth count, but not the diameters.

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in teeth | Number of teeth | - | 20 | 5
in module | Module (metric) | mm | 2 | 0.1
out pitch_d | Pitch diameter | mm | teeth * module | 2
out outer_d | Outer diameter | mm | (teeth + 2) * module | 2
out root_d | Root diameter | mm | (teeth - 2.5) * module | 2