Calculate angular displacement (θ) using three different methods depending on what values you know. Enter an arc length and radius, or provide angular velocity and time, or use initial angular velocity, angular acceleration, and time — and get the angular displacement in both radians and degrees.
Angular Displacement (θ)
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Angular Displacement in Degrees
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Number of Revolutions
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Formula Used
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Angular displacement is the angle through which an object moves along a circular path. It is measured in radians (or degrees) and represents how far the object has rotated from its initial position. Unlike linear displacement, it describes rotational motion and is a vector quantity, meaning it has both magnitude and direction.
There are three common formulas: (1) θ = s / r, where s is arc length and r is the radius; (2) θ = ω × t, where ω is angular velocity and t is time; and (3) θ = ω₀t + ½αt², where ω₀ is initial angular velocity, α is angular acceleration, and t is time. Choose the formula based on which quantities you already know.
The standard SI unit for angular displacement is the radian (rad). It can also be expressed in degrees (°) or revolutions (rev). One full revolution equals 2π radians or 360°.
If the angular velocity (ω) is constant, angular displacement is calculated as θ = ω × t, where t is the time elapsed. For example, if an object spins at 5 rad/s for 4 seconds, its angular displacement is 20 radians.
When angular acceleration (α) is involved, use the kinematic equation θ = ω₀t + ½αt², where ω₀ is the initial angular velocity and t is the elapsed time. This is the rotational analogue of the linear kinematic equation s = u·t + ½a·t².
One full rotation equals 2π radians, so three full rotations give an angular displacement of 3 × 2π ≈ 18.8496 radians, which equals 1080°.
Linear displacement measures how far an object has moved in a straight line, while angular displacement measures the angle swept during rotational motion. They are related by the formula s = r × θ, where s is arc length (linear distance along the curve) and r is the radius of the circular path.
Angular displacement is a vector quantity — it has both magnitude (the angle) and direction (clockwise or counterclockwise, often defined using the right-hand rule). However, for small angular displacements, it can be treated as a scalar in many practical calculations.