What this calculator does
The involute function is short: inv α = tan α − α, with α in radians. Its importance is out of all proportion to its length, because the involute curve is the tooth profile almost every gear in service uses, and this function is what relates an angle on that curve to the position along it.
The one thing to be careful of is units. The tangent is dimensionless but α has to be subtracted from it in radians, not degrees, or the result is meaningless. This calculator takes the pressure angle in degrees because that is how gear specifications are written, converts internally, and returns the involute in both radians and degrees so either convention can be read off.
The formula
The pressure angle entered in degrees is converted to radians. Its tangent is taken, the angle itself in radians is subtracted from that tangent, and the difference is the involute. The result is reported in radians, which is the form gear formulas expect, and converted back to degrees alongside for anyone working from a degree-based table.
| Term | Meaning |
|---|---|
| inv α | The involute function of α, equal to tan α − α with α in radians. |
| Pressure angle | The angle between the line of action and the common tangent of two meshing gears, commonly 20 degrees on modern gearing. |
| Involute curve | The path traced by the end of a taut string unwound from a circle, and the profile of a standard gear tooth. |
| Base circle | The circle the involute is generated from, which fixes the whole tooth profile once the pressure angle is chosen. |
The inputs explained
| Field | What to enter |
|---|---|
| Pressure angle α (degrees) | The pressure angle in degrees. It must sit strictly between −90 and 90, since the tangent is unbounded at either end. |
When to use it
Calculating gear tooth thickness at a given radius
The standard formula for tooth thickness away from the pitch circle is written in terms of the involute of the pressure angle at that radius and at the reference radius. Both values come from this function.
Working out a span measurement over teeth
Measuring over several teeth with a gauge is the usual practical check on gear geometry, and the expected span measurement is derived directly from the involute of the pressure angle.
Checking a value against a printed involute table
Gear handbooks carry tables of inv α to five or six decimal places. Computing the value directly confirms a table reading, particularly for the angles that fall between printed entries.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is the involute of the standard pressure angles?
The involute function evaluated at the pressure angles that appear on gear specifications, plus 45 degrees for scale.
| Pressure angle α | Involute φ (radians) | Involute φ (degrees) |
|---|---|---|
| 14.5° | 0.005545 | 0.3177 |
| 20° | 0.014904 | 0.8540 |
| 25° | 0.029975 | 1.717 |
| 30° | 0.053751 | 3.080 |
| 45° | 0.214602 | 12.296 |
Questions
Does α need to be in radians or degrees?
Radians, in the formula itself. tan α − α only makes sense if the angle being subtracted is in the same natural units the tangent implicitly uses. This calculator accepts degrees for convenience, because that is how pressure angles are specified, and converts before computing.
Why is 20 degrees the standard pressure angle?
It is a compromise that settled through use. Lower angles such as 14.5 degrees give smoother, quieter running but weaker teeth with more undercutting on small gears; higher angles such as 25 degrees give stronger teeth but more separating force on the bearings. 20 degrees sits between the two and became the default.
Can the involute function be inverted?
Not in closed form. Recovering α from a known inv α requires numerical iteration, usually Newton-Raphson, which converges quickly since the function is smooth and strictly increasing over the working range.
Why does the involute grow so much faster than the angle?
Because the tangent does. Near zero, tan α and α are almost identical and their difference is tiny, growing roughly as the cube of the angle. As α climbs towards 90 degrees the tangent runs away to infinity while α does not, so the gap widens without limit.
For the trigonometric functions the involute is built from, see the trigonometry calculator. To work with angles and their conversions directly, see the angle and trigonometry calculator.