# Involute Function Calculator

URL: https://speedycalculator.com/calculators/math/involute-function/
Last updated: 2026-10-30

> The involute function calculator evaluates inv α = tan α − α, used in gear design for involute tooth profiles, for any pressure angle in degrees or radians.

## Inputs

- Angle α (°, rad, π rad, grad, turn)

## Results

- inv α
- tan α
- α in radians

## Formula

```
inv α = tan α − α  (α in radians)
```

## Explanation

Almost all modern gears have involute tooth profiles, which keep the speed ratio constant as teeth roll together. The involute function converts pressure angles into tooth thickness and centre distance calculations.

### How to use the Involute Function Calculator

- **Angle α**

Results update as you type, and everything is calculated in your browser.

### How it works

Enter the angle (the common pressure angle is 20°). The function is computed with α in radians.

### Tips

- Read the short explanation under each result – these tools illustrate classic ideas rather than everyday sums.
- Change one input at a time to see how the result reacts.
- Everything runs in your browser, so you can experiment freely.
- Check the worked example below the calculator if a result surprises you.

### Common mistakes to avoid

- Using the wrong units for rates in queueing models.
- Expecting paradoxes to follow everyday intuition.
- Entering text where a number is expected.

### What each result means

- **inv α** = tan(rad([Angle α])) - rad([Angle α])
- **tan α** = tan(rad([Angle α]))
- **α in radians** = rad([Angle α])

### Example values

How the main result changes when **Angle α** changes and everything else stays at the example values above:

Angle α (°)inv α100.0018150.0061200.0149250.03300.0538400.141

## Worked example

Inputs — Angle α: 20 °.

Results — **inv α = 0.014904383867**; tan α = 0.363970234266; α in radians = 0.349065850399.

## Key terms

- **Involute**: The curve traced by unwinding a taut string from a circle.

## FAQ

**What is the involute of 20°?**
About 0.014904.

**What is an involute curve?**
The path traced by the end of a string unwound from a circle.

**Why are gear teeth involutes?**
They keep a constant velocity ratio even if the centre distance varies slightly.

**What is a pressure angle?**
The angle between the tooth force and the pitch-circle tangent, usually 20°.

**What formula does the Involute Function Calculator use?**
inv α = tan α − α  (α in radians).


## Sources

- [Wolfram MathWorld – Circle involute](https://mathworld.wolfram.com/CircleInvolute.html)