Angular Displacement Calculator

Created by Dominik Czernia, PhD
Last updated: Jul 01, 2022

Our angular displacement calculator consists of three methods of estimating the angular displacement formula depending on the known parameters. Some require other physical quantities related to rotational motion, i.e., angular velocity and angular acceleration. You can read more about them in the dedicated tools, angular velocity calculator and angular acceleration calculator, respectively.

In the below article, we focus on explaining what angular displacement is, how to find angular displacement, and what is a unit for angular displacement. Additionally, you will find an answer to the following example:

If a wheel rotates 3 times, what is its angular displacement?

Angular displacement formula

Let's start by explaining the available angular displacement equations. By definition, angular displacement is the angle by which a body rotates around a specified axis. Following that sentence, you can immediately write one of the formulae that we use in the angular displacement calculator:

$\theta = \omega \times t$

where:

• $\theta$ - The angular displacement (the units of angular displacement are radians or degrees);
• $\omega$ - The angular velocity; and
• $t$ - Time passed.

The second way of showing how to calculate angular displacement assumes an object is in a circular motion. You can check how to calculate circular motion properties in one of our other calculators. If the radius of the circular path is $r$ and the object traveled the distance of $s$ on the circumference, then the angular displacement equation is the following:

$\theta = \frac{s}{r}$

where you get the final result in radians. Of course, the angular displacement calculator can automatically switch the units to degrees if you prefer. Finally, let's write the angular displacement formula using angular acceleration $\alpha$:

$\theta = \omega_0 \times t + \frac{1}{2} \times \alpha \times t^2$

where, this time, $\omega_0$ is the initial angular velocity.

If a wheel rotates 3 times, what is its angular displacement?

The above question isn't about how to calculate angular displacement but rather how to switch units appropriately. You can still do it with our tool:

1. The angular displacement is simply three rotations, but you might want to use different units.
2. To do this, select any formula in the angular displacement calculator.
3. One rotation is 360 degrees, so select the degrees as units in the angular displacement field and type in 360*3 which is 1080 degrees, by simple multiplication. That's one of the possible results.
4. If you want the result in radians, change the units appropriately, and you will get 18.85 radians.

That's it! If you want to learn something new after obtaining your answer, visit the arcsin calculator to explore this trigonometric function that also involves angles.

Dominik Czernia, PhD
Formula:
Radius of circular path is known
Distance travelled (s)
m
Radius of the circular path (r)
m
Angular displacement (θ)