Angular Velocity Calculator
Enter a rotational frequency and a radius to get the angular velocity in radians per second — plus the tangential speed — and see how fast a spinning object really turns.
Angular and tangential at once
Enter the frequency and radius and the calculator returns the angular velocity (2π·f) in rad/s and the tangential speed (ω·r) in m/s together.
Use Hz and metres
Frequency in hertz (turns per second) and radius in metres give the answer in rad/s and m/s — divide rpm by 60 to get hertz before you start.
What is angular velocity?
The speed of rotation
The angular velocity calculator turns a rotational frequency into the rate at which an object sweeps through angle, measured in radians per second. Angular velocity describes how fast something spins regardless of its size: a record player, a car wheel, and a turbine all have a well-defined angular velocity. From the frequency in hertz it also derives the tangential speed — how fast a point at a chosen radius actually travels along its circular path. It is the number behind wheel speeds, motor ratings, and the design of anything that turns.
Enter a frequency in hertz and a radius in metres to get the angular velocity in rad/s and the tangential speed in m/s instantly.
Angular velocity is two pi multiplied by the rotational frequency, and the tangential speed is that angular velocity multiplied by the radius.
ω = 2π × fOne full revolution is 2π radians, so multiplying the frequency in hertz by 2π converts turns per second into radians per second. The tangential speed (v = ω × r) then scales with the radius: the further a point sits from the axis, the faster it moves. Use hertz and metres and the answers come back in rad/s and m/s.
Suppose a disc spins once every second (1 Hz) and we look at a point 0.5 m from the centre.
Multiply the frequency by 2π
2π × 1 = 6.283185 rad/s — the angular velocity of the whole disc.
Multiply by the radius
6.283185 × 0.5 = 3.141593 — angular velocity times the radius.
Read the tangential speed
The point 0.5 m out moves at 3.141593 m/s along its circular path, while the disc itself turns at 6.283185 rad/s.
The two outputs answer two different questions. The angular velocity (6.283185 rad/s above) is a property of the whole rotating body — every point shares it, from the hub to the rim. The tangential speed (3.141593 m/s) belongs to one specific point and depends on how far that point sits from the axis. Double the radius and the tangential speed doubles, even though the angular velocity is unchanged; that is why the rim of a wheel moves much faster than a point near its centre. This relationship explains why long blades, large rotors, and the outer edges of spinning machinery reach high speeds at modest rotation rates, and why frequency and radius must both be considered when judging whether a moving part is safe or efficient.
The formula is exact, but a couple of practical points are worth keeping in mind.
Constant rotation and consistent units
This calculator assumes steady rotation at a single frequency and gives the magnitude of the angular velocity, not its direction. It does not cover angular acceleration (speeding up or slowing down). Keep your units consistent — hertz for frequency and metres for radius — or the result will be wrong: convert rpm to hertz by dividing by 60 before you enter the frequency.