Involute Gear Geometry

Involute spur gears look complex, but the first-pass geometry is controlled by a short list of inputs: tooth size, tooth count, pressure angle and profile shift. Once those are fixed, the pitch diameter, base circle, outside diameter, root diameter, pitch and tooth thickness all follow from standard formulas.

The core inputs

Symbol	Meaning	Typical value
`m`	Module, the metric tooth-size unit	1, 1.5, 2, 2.5, 3 mm...
`DP`	Diametral pitch, the imperial tooth-size unit	8, 10, 12, 16, 20...
`z`	Tooth count	Application dependent
`α`	Pressure angle	20° modern standard
`x`	Profile-shift coefficient	0 for an unshifted standard gear

Module and diametral pitch describe the same physical tooth size in opposite unit systems. Convert with:

m = 25.4 / DP DP = 25.4 / m

The module vs diametral pitch guide covers that conversion in more detail.

Pitch diameter and circular pitch

The reference pitch diameter is the basic size of the gear:

d = m · z

The circular pitch is the arc distance from one tooth to the next on the pitch circle:

p = π · m

Two mating spur gears must have the same module, or the same diametral pitch, so their tooth spacing matches.

Base circle and involute flank

An involute tooth flank is generated from the base circle. The base circle diameter is tied to pressure angle:

d_b = d · cos α

The base pitch is the pitch projected onto the base circle:

p_b = p · cos α

This is why mating gears need the same pressure angle as well as the same tooth size: both the spacing and the flank shape must agree.

Addendum, dedendum and diameters

For standard full-depth teeth with no profile shift, the addendum is one module and the dedendum is 1.25 modules:

a = m

b = 1.25m

Therefore:

d_a = d + 2m = m(z + 2)

d_f = d − 2.5m = m(z − 2.5)

With profile shift x, the calculator uses:

a = m(1 + x) b = m(1.25 − x)

d_a = d + 2a d_f = d − 2b

Tooth thickness and backlash

For an unshifted standard gear, circular tooth thickness at the pitch circle is half the circular pitch:

s = p / 2 = πm / 2

With profile shift, the pitch-circle tooth thickness becomes:

s = m(π/2 + 2x tan α)

Real gears need backlash. A quick early allowance is roughly 0.03m to 0.05m, but production backlash is set by tooth-thickness tolerance, center-distance tolerance and the gear quality class.

Center distance

Two unshifted external spur gears with tooth counts z₁ and z₂ have this standard center distance:

a₀ = m(z₁ + z₂) / 2

A profile-shifted pair uses a working pressure angle and a modified center distance. That is where the involute function appears:

inv α = tan α − α

For practical use, let the involute gear calculator solve the shifted case and export the tooth profile as DXF.

Undercut check

Small pinions can be undercut by the generating cutter near the root. A common full-depth approximation for the minimum tooth count without undercut is:

z_min = 2 / sin² α

At α = 20°, this gives z_min ≈ 17.1, so a standard 20° pinion below about 17 teeth needs attention. Options include more teeth, a positive profile shift, a larger pressure angle, or a stub tooth system.

Worked example

Take a module-2, 20° spur gear with z = 20 teeth and no profile shift.

Pitch diameter: d = 2 × 20 = 40 mm
Base diameter: d_b = 40 × cos 20° = 37.59 mm
Circular pitch: p = π × 2 = 6.28 mm
Outside diameter: d_a = 2(20 + 2) = 44 mm
Root diameter: d_f = 2(20 − 2.5) = 35 mm
Tooth thickness: s = π × 2 / 2 = 3.14 mm

If it meshes with a 40-tooth gear of the same module and pressure angle, the center distance is 2(20 + 40)/2 = 60 mm.

References

Frequently asked questions

What geometry defines an involute spur gear?

A standard involute spur gear is mainly defined by module or diametral pitch, tooth count, pressure angle and profile shift. Those determine pitch diameter, base diameter, addendum, dedendum, outside diameter, root diameter and tooth thickness.

What is the base circle of a gear?

The base circle is the circle from which the involute tooth flank is generated. For a spur gear, d_b = d × cos α, where d is pitch diameter and α is pressure angle.

Why are 20 degree gears common?

A 20 degree pressure angle is the modern standard compromise: less undercut than 14.5 degree teeth and less separating force than 25 degree teeth. Mating gears must share the same module or DP and pressure angle.

How many teeth avoid undercut?

For standard full-depth involute teeth, the approximate minimum tooth count is z_min = 2 / sin²α. At 20 degrees that is about 17 teeth. Positive profile shift can reduce undercut risk on smaller pinions.

Ready to run the numbers?

Open the Involute Gear Calculator

Last reviewed: 2026-05-30.