What is angular displacement?

Angular displacement is the angle through which a point or line has been rotated in a specified direction about a specified axis. It is measured in radians (or degrees) and can be positive (counterclockwise) or negative (clockwise).

What is the formula for angular displacement?

The kinematic formula for angular displacement is: Δθ = θ - θ₀ = ω·t + ½·α·t², where θ is the final angular position, θ₀ is the initial angle, ω is the initial angular velocity, α is the angular acceleration, and t is time. The arc-length formula is Δθ = s / r, where s is the arc length and r is the radius.

What units should I use?

Use radians for angular quantities (angle, displacement). The calculator also accepts degrees — just select the appropriate unit. Angular velocity should be in rad/s and angular acceleration in rad/s^2 when using SI units.

What is the difference between angular displacement and angle?

An angle describes a geometric position, while angular displacement describes the change in angle from an initial to a final position. Angular displacement can exceed 2*pi (one full revolution) and carries a sign indicating direction.

Can angular displacement be negative?

Yes. A negative angular displacement means the object rotated in the clockwise direction (by convention in standard 2-D analysis where counterclockwise is positive).

Angular Displacement Calculator

Enter angular velocity and time (or arc length and radius) to instantly calculate angular displacement in radians and degrees.

Calculation Mode

Initial Angle θ₀

Angular Velocity ω (rad/s)

Angular Acceleration α (rad/s²) (optional, default 0)

Time t (seconds)

Formula Reference

Δθ = θ - θ₀ = ω·t + ½·α·t²

Δθ — angular displacement (rad)
θ — final angular position (rad)
θ₀ — initial angle (rad)
ω — initial angular velocity (rad/s)
α — angular acceleration (rad/s²)
t — time (s)

Enter values on the left and click Calculate

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Summary

Angular displacement measures how far an object has rotated about a fixed axis, expressed in radians or degrees. This calculator supports two common methods: the kinematic formula (Δθ = θ - θ₀ = ω·t + ½·α·t²) and the arc-length method (Δθ = s/r). Enter your known values and get the result instantly without any server round-trip.

How it works

Choose a calculation mode: Kinematic (angular velocity + time) or Arc Length (arc length + radius).
In Kinematic mode, enter the initial angle, angular velocity (omega), angular acceleration (alpha, optional), and elapsed time.
In Arc Length mode, enter the arc length traveled and the radius of the circular path.
Click Calculate to compute the angular displacement (Δθ).
Results are shown in both radians and degrees for easy reference.

Use cases

Solving rotational kinematics problems in physics coursework.
Calculating how many radians a wheel or gear rotates in a given time.
Converting arc length on a circular path to the corresponding central angle.
Verifying manual calculations for engineering and robotics projects.
Determining the rotation angle of a motor shaft at a given speed and time.
Classroom demonstrations of angular displacement concepts.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu