Angular Displacement Calculator
Choose a formula mode - arc length and radius, angular velocity and time, or the full kinematics equation with angular acceleration - then enter your values. Angular displacement, in radians, degrees, and full rotations, updates instantly. A step-by-step panel shows the working with your numbers.
Formula
Worked example
A wheel with a 2 m radius rolls 6 m along the ground: theta = 6 / 2 = 3 rad = 171.9 deg = 0.477 rotations. Alternatively, if the wheel spins at 3 rad/s for 4 s: theta = 3 x 4 = 12 rad = 687.5 deg = 1.91 rotations. Under an angular acceleration of 1 rad/s^2 from 2 rad/s for 3 s: theta = 2(3) + 0.5(1)(3^2) = 6 + 4.5 = 10.5 rad.
What is angular displacement?
Angular displacement is the signed angle through which a point, line, or body rotates about a fixed axis between two instants in time. It is measured in radians (the SI unit), degrees, or full rotations (revolutions). Unlike linear displacement, which is a straight-line distance between two positions, angular displacement describes how far a rotating object has turned. A positive value conventionally means counterclockwise rotation; a negative value means clockwise. When a body completes more than one full turn, angular displacement can exceed 2 pi radians (360 degrees), whereas the position angle simply wraps back to zero.
Which formula should I use?
The right formula depends on what you know. If you know how far a point on a rotating object has traveled along its circular path (the arc length, s) and the radius (r) of that path, use theta = s / r. This is the definition of the radian and the most direct relationship between linear and angular motion. If you know the constant angular velocity (omega, in rad/s) and the time elapsed, use theta = omega x t. If the angular velocity is changing because there is an angular acceleration (alpha, in rad/s^2), use the kinematic equation theta = omega_0 x t + 0.5 x alpha x t^2, where omega_0 is the initial angular velocity. This is the rotational analogue of the linear equation d = v_0 t + 0.5 a t^2.
Radians, degrees, and rotations explained
A radian is the angle subtended at the center of a circle by an arc equal in length to the radius. There are exactly 2 pi radians in one full rotation (360 degrees). To convert radians to degrees, multiply by 180/pi (about 57.296). To convert to rotations, divide by 2 pi. Radians are preferred in physics and engineering because they make formulas cleaner: arc length is simply s = r x theta when theta is in radians, with no conversion factor needed. Degrees are convenient for everyday descriptions of angles. Rotations (also called revolutions) are most useful when counting how many full circles an object has completed, such as in gear ratios or motor specifications.
Angular displacement in kinematics
Rotational kinematics mirrors linear kinematics with a one-to-one substitution of variables: displacement theta replaces position x, angular velocity omega replaces linear velocity v, and angular acceleration alpha replaces linear acceleration a. The four kinematic equations all have rotational counterparts. This calculator solves the position equation. The others are: omega = omega_0 + alpha x t (final angular velocity), omega^2 = omega_0^2 + 2 x alpha x theta (velocity from displacement), and theta = 0.5 x (omega_0 + omega) x t (average velocity). These equations assume constant angular acceleration, which covers most introductory physics problems involving motors, wheels, pulleys, and spinning bodies.
Common angular displacement reference values
| Description | Radians | Degrees | Rotations |
|---|---|---|---|
| Quarter turn | pi / 2 (1.5708) | 90 deg | 0.25 rev |
| Half turn | pi (3.1416) | 180 deg | 0.5 rev |
| Full rotation | 2 pi (6.2832) | 360 deg | 1 rev |
| Clock minute hand (1 min) | pi / 30 (0.1047) | 2 deg | 0.00278 rev |
| Clock hour hand (1 hr) | pi / 6 (0.5236) | 30 deg | 0.0833 rev |
| Wheel at 33 RPM (1 s) | 3.456 | 198 deg | 0.55 rev |
Familiar angles and their equivalents in radians, degrees, and rotations.
Frequently asked questions
What is the difference between angular displacement and angle?
Angle usually refers to the geometric measure between two rays at a given instant, bounded between 0 and 2 pi. Angular displacement is a physical quantity describing how much an object has rotated over a time interval. It is cumulative and signed: a body that completes 3 full turns has an angular displacement of 6 pi radians, not zero, and a clockwise rotation is negative by convention.
How do I convert radians to degrees?
Multiply the radian value by 180/pi (approximately 57.2958). For example, pi/2 radians x (180/pi) = 90 degrees. To go the other way, multiply degrees by pi/180.
Can angular displacement be negative?
Yes. By the standard mathematical convention, counterclockwise rotation is positive and clockwise rotation is negative. A negative angular displacement simply means the object turned in the clockwise direction. The magnitude of the displacement is the same; only the direction differs.
What is the formula for angular displacement using arc length?
theta = s / r, where s is the arc length (distance traveled along the circular path) and r is the radius of the circle. The result is in radians as long as s and r are in the same length unit. This formula comes directly from the definition of the radian.
What is the kinematic formula for angular displacement?
When angular acceleration is constant, theta = omega_0 x t + 0.5 x alpha x t^2, where omega_0 is the initial angular velocity in rad/s, alpha is the angular acceleration in rad/s^2, and t is time in seconds. If alpha is zero (constant speed), this simplifies to theta = omega x t.
How many radians are in a full rotation?
2 pi radians (approximately 6.2832 rad) equal one full rotation (360 degrees). One radian is about 57.3 degrees.