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Spur Gear Dimension Formulas

Guide · Updated October 28, 2026

Before cutting a spur gear you need to turn the blank to the right outside diameter, know how deep to cut, and know where to put the shafts. All of these come from the tooth count and the tooth size, given as diametral pitch (DP, teeth per inch of pitch diameter) for inch gears or module (m, mm of pitch diameter per tooth) for metric gears. The formulas below are for standard full-depth involute spur gears with 14.5° or 20° pressure angle.

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Step by step

Get tooth count and pitch: From the drawing or the mating gear: N teeth and DP, or z teeth and module m.
Find pitch diameter: PD = N ÷ DP, or d = m × z.
Turn the blank OD: OD = (N + 2) ÷ DP, or da = m × (z + 2).
Set the depth: Whole depth about 2.25 ÷ DP, or 2.25 × m (fine pitch 20 DP and finer: 2.2 ÷ DP + 0.002 in).
Set center distance: C = (N1 + N2) ÷ (2 × DP), or (z1 + z2) × m ÷ 2.
Check the gear: Measure over pins or with a gear tooth vernier, and check the blank with a mic.

Inch (diametral pitch) formulas

Pitch diameter PD = N ÷ DP
Outside diameter OD = (N + 2) ÷ DP
Addendum a = 1 ÷ DP
Dedendum b = 1.25 ÷ DP (coarse pitch)
Whole depth ht = 2.25 ÷ DP (coarse), 2.2 ÷ DP + 0.002 (fine, 20 DP and finer)
Circular pitch p = π ÷ DP
Tooth thickness at pitch line t = 1.5708 ÷ DP
Center distance C = (N1 + N2) ÷ (2 × DP)

Metric (module) formulas

Pitch diameter d = m × z
Tip diameter da = m × (z + 2)
Addendum = m, dedendum = 1.25 × m
Whole depth h = 2.25 × m
Circular pitch p = π × m
Center distance a = m × (z1 + z2) ÷ 2
Conversion: DP = 25.4 ÷ m

Worked example: 20 DP, 40 teeth

PD = 40 ÷ 20 = 2.000 in. OD = 42 ÷ 20 = 2.100 in. Whole depth for fine pitch = 2.2 ÷ 20 + 0.002 = 0.112 in (the coarse-pitch rule gives 0.1125 in, almost the same). Circular pitch = 0.1571 in and tooth thickness = 0.0785 in. Meshing with a 20 tooth 20 DP pinion, center distance = (40 + 20) ÷ 40 = 1.500 in.

Worked example: module 2, 30 teeth

d = 2 × 30 = 60 mm. da = 2 × 32 = 64 mm. Whole depth = 4.5 mm. Circular pitch = 6.283 mm. With a 15 tooth mate, a = 2 × 45 ÷ 2 = 45 mm. Module 2 is about 12.7 DP, so it will not mesh with a 12 DP gear.

Picking a cutter

Involute gear cutters in the common 8-cutter series cover a range of tooth counts per cutter: No. 1 for 135 teeth to a rack, No. 2 for 55 to 134, No. 3 for 35 to 54, No. 4 for 26 to 34, No. 5 for 21 to 25, No. 6 for 17 to 20, No. 7 for 14 to 16 and No. 8 for 12 to 13. A 40 tooth gear uses a No. 3 cutter. Match the pressure angle (14.5° or 20°) of the mating gear.

Common problems

Turning the blank to pitch diameter instead of OD, mixing 14.5° and 20° gears, cutting full depth on an oversize blank, and confusing module with DP. Mark the blank OD and depth on the setup sheet.

FAQ

What is the OD of a gear?

(N + 2) ÷ DP for inch gears, or m × (z + 2) for metric.

How deep do I cut a gear tooth?

About 2.25 ÷ DP or 2.25 × module for full-depth teeth.

How do I convert module to DP?

DP = 25.4 ÷ module.

How do I check the gear blank OD?

Check the gage itself before trusting a measurement: gage blocks set mics, calipers and indicators; build any stack free at gageblockcalc.com.

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General shop information only — follow your drawing, tool maker's data and your quality system.