Involute Function Calculator

Background.

Most gear teeth, in car gearboxes, power tools and industrial drives, are shaped as the involute of a circle: the curve traced by the end of a taut string as it is unwound from a cylinder called the base circle. Engineers favour it because two involute teeth always push on each other along one fixed straight line, so a pair of gears keeps a constant speed ratio even when the distance between their centres is slightly off.

Work with that curve keeps producing one small expression, tan α − α, written inv α and called the involute function. Here α is the pressure angle at a point on the tooth, the angle whose cosine is the base-circle radius divided by the point's distance from the gear centre. Geometrically, inv α is how far round, in radians, the point sits from where the involute leaves the base circle. A point with a 20° pressure angle is 0.014904 radians round, a little under one degree.

That angular offset is exactly what gear calculations need. Tooth thickness at any diameter, the measurement over pins or balls used to inspect a finished gear, and the working pressure angle of a pair with shifted tooth profiles are all written in terms of inv α, and before calculators engineers looked the values up in printed tables.

Enter an angle in degrees, anywhere from −89.9° to 89.9°; the page converts it to radians and returns tan α − α in radians. Negative angles give the mirror-image value because the function is odd. The reverse problem, finding the angle that has a given involute value, is not solved here, and neither is choosing a pressure angle or designing a tooth.

What is involute function calculator?

The involute function is inv α = tan α − α, with α in radians. In gear geometry it is the polar angle of a point on an involute curve, measured at the gear centre from the point where the curve starts on the base circle, and α is the pressure angle at that point. It is zero at α = 0, grows roughly as α³/3 for small angles, and rises without limit as α approaches 90°, where the tangent becomes infinite.

How to use this calculator.

1. 01Enter the pressure angle in degrees. At the pitch circle of a standard gear that is the nominal pressure angle, most often 20°, with 14.5° on many older designs and 25° on some heavily loaded gears.
2. 02For a point at some other radius r on the tooth, first find its own pressure angle from cos α = r_b ÷ r, where r_b is the base-circle radius, and enter that angle instead.
3. 03Read inv α in radians and keep every digit shown. Involute values for working angles are small, and in tooth-thickness formulas they are multiplied by a radius, so trimming decimals shows up as a thickness error.
4. 04Carry the value into the tooth-thickness, measurement-over-pins or operating-pressure-angle formula you are using.

The formula.

inv α = tan α − α. The page converts the angle with α = degrees × π ÷ 180, takes its tangent and subtracts the angle itself. Radians are essential, because tan α is a pure ratio and only an angle defined as arc length over radius can be subtracted from it. The expression falls straight out of the involute's construction. When the string has been unwound to a point whose pressure angle is α, its free length is r_b tan α, which equals the arc it has left on the base circle, so the total roll angle is tan α. Of that roll, α is the angle at the centre between the string's tangent point and the point itself, leaving tan α − α as the point's polar angle. Since the tangent grows faster than the angle, the function climbs steeply: moving from 20° to 25° roughly doubles it.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.