Angular displacement is the angle through which something rotates. It is measured in radians, where one full turn is 2π rad. It is used for wheels, gears, rotating machinery and planets.
How to use the Angular Displacement Calculator
- What do I know?
- Distance along the arc s
- Radius r
- Angular velocity ω
- Time t
- Angular acceleration α
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose what you know. The result is given in radians, degrees and revolutions.
Inputs explained
| Field | What to enter |
|---|---|
| What do I know? | Choose one: Arc length and radius, Angular velocity and time, Initial angular velocity, acceleration and time |
| Distance along the arc s | Units: m, cm, mm, km, in, ft … |
| Radius r | Units: m, cm, mm, km, in, ft … |
| Angular velocity ω | Units: rad/s, rpm, Hz (rev/s), °/s |
| Time t | Units: s, ms, min, h, days |
| Angular acceleration α | Units: rad/s², °/s², rpm/s |
What each result means
- In degrees = [In degrees]([Angular displacement θ])
- In revolutions = [Angular displacement θ]÷(2×pi)
Example values
How the main result changes when Distance along the arc s changes and everything else stays at the example values above:
| Distance along the arc s (m) | Angular displacement θ (rad) |
|---|---|
| 1.5 | 3 |
| 2.25 | 4.5 |
| 3 | 6 |
| 3.75 | 7.5 |
| 4.5 | 9 |
| 6 | 12 |
Formula
θ = s/r · θ = ω·t · θ = ω₀·t + ½·α·t²
Worked example
Inputs — What do I know?: Arc length and radius; Distance along the arc s: 3 m; Radius r: 0.5 m.
Results — Angular displacement θ = 6 rad; In degrees = 343.7747 °; In revolutions = 0.95493 rev.
Key terms
- Angular displacement
- The angle rotated, in radians.
Frequently asked questions
What is a radian?
The angle for which the arc length equals the radius; 2π rad = 360°.
Is angular displacement a vector?
For small rotations it is treated as a vector along the axis.
How far does a wheel roll after θ radians?
s = r·θ.
Can it exceed 2π?
Yes – it counts total rotation, including full turns.