Involute Function Calculator (inv α = tan α − α and its inverse, for gears)

Understanding the involute function calculator — concept and principle

This calculator evaluates the involute function inv α = tan α − α used in gear geometry, or finds the angle α for a given involute value. It shows α in degrees and radians, tan α and inv α expressed in degrees. Use it for gear tooth thickness, profile-shifted gears and checking values from involute tables.

The involute function is the gap between the tangent and the arc of the same angle.

How it is calculated

inv α = tan α − α, with α in radians

Radians = degrees × π ÷ 180.
The inverse (α from inv α) has no closed form and is solved iteratively, for example with Newton's method.
Typical gear uses: tooth thickness on another circle, s = d(s₀/d₀ + inv α₀ − inv α), and the working pressure angle of profile-shifted gears, inv α_w = 2 tan α₀ (x₁ + x₂) ÷ (z₁ + z₂) + inv α₀.

The principle

Unwind a taut string from a circle and its end traces an involute, the standard profile of gear teeth. At a point where the profile makes pressure angle α with the radius, the unwound string length is r_b·tan α (r_b = base-circle radius), so the angle unwound around the base circle is tan α. Subtracting the angle α itself leaves the polar angle of that point measured from the start of the involute: tan α − α. Involute teeth keep a constant pressure angle and a constant speed ratio even if the centre distance changes slightly.

Worked example

Default inputs: α = 20°, calculate inv α.

α in radians = 20 × π ÷ 180 = 0.349066.
tan 20° = 0.36397.
inv 20° = 0.36397 − 0.349066 = 0.014904384 (shown as 0.0149044).
Expressed in degrees: 0.0149044 × 180 ÷ π = 0.853958°.
Method: direct: tan α − α.

Each point is the calculator re-run with only “Angle α in degrees (first option)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Angle α in degrees (first option)	Answer
5	0.0002222009
10	0.0017940555
15	0.0061498046
20	0.014904384
25	0.029975345
30	0.053751494
40	0.14096793

How to read the result

20° is the most common standard pressure angle, so inv 20° ≈ 0.014904 appears in many gear calculations. The involute function grows quickly with α: small angles give tiny values, so keep enough digits; rounding inv α to 0.015 already changes tooth-thickness results noticeably.

Common mistakes

Subtracting α in degrees from tan α.
Rounding inv α too early.
Using the pressure angle at the pitch circle when another circle is meant.
Expecting an exact formula for the inverse.

Keep in mind: Gear design involves many more factors, such as backlash, tolerances and standards for tooth form. Real gears need design by qualified engineers to the applicable standards.

Frequently asked questions

What is the involute function?

inv α = tan α − α (radians), the polar angle of a point on a circle involute at pressure angle α.

What is inv 20°?

About 0.014904.

Why are gear teeth involute-shaped?

Involute teeth transmit motion at a constant speed ratio and tolerate small changes in centre distance.

Sources: tec-science, "Involute function" (gear geometry)

Checked 9 Nov 2026. tec-science, “Involute function” (inv α = tan α − α with all angles in radians; use in tooth thickness s = d·(s₀/d₀ + inv α₀ − inv α) and in the operating pressure angle inv α_b = 2(x₁ + x₂)/(z₁ + z₂)·tan α₀ + inv α₀; no closed-form inverse, solved iteratively, e.g. with Newton’s method). Standard involute gear geometry.

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