Angular Frequency Calculator
Enter an ordinary frequency in hertz to get the angular frequency ω = 2πf in radians per second — and see why one cycle per second equals 2π rad/s.
One field, one answer
Enter the frequency in hertz and this angular frequency calculator returns the angular frequency ω (omega) in radians per second using ω = 2πf.
Use hertz
The frequency must be in hertz (cycles per second). One hertz equals 2π radians per second, because one full cycle sweeps 2π radians.
What is angular frequency?
Radians per second instead of cycles per second
Angular frequency is how fast something oscillates or rotates, expressed in radians per second rather than cycles per second. Where the ordinary frequency f counts how many full cycles happen each second (in hertz), the angular frequency ω measures the same motion as an angle swept per second. Since one full cycle is 2π radians, the two are linked by ω = 2πf. This angular frequency calculator turns a frequency in hertz into ω in radians per second — the form physicists use in the equations for waves, oscillations, alternating current, and circular motion.
Enter a frequency in hertz to get the angular frequency in radians per second instantly.
The angular frequency is simply two pi multiplied by the ordinary frequency.
ω = 2 × π × fThe factor 2π converts cycles into radians: one complete cycle corresponds to 2π radians, so multiplying the frequency in hertz by 2π gives the rate in radians per second. Put a frequency in hertz in, and ω comes back in radians per second.
Suppose a signal oscillates at 50 Hz (the European mains frequency).
Start with the frequency
f = 50 Hz — fifty full cycles every second.
Multiply by 2π
2 × π = 6.283185 — the radians in one full cycle.
Scale by the frequency
6.283185 × 50 = 314.16 rad/s — the angular frequency. The signal sweeps about 314.16 radians every second.
The result is an angular rate measured in radians per second, not cycles per second. The two describe the same motion from different angles: the ordinary frequency f counts whole cycles per second (hertz), while the angular frequency ω counts the angle, in radians, swept in that same second. Because one cycle is 2π radians, 1 Hz always equals 2π ≈ 6.2832 rad/s, and the 50 Hz example above works out to about 314.16 rad/s. Angular frequency is the natural form for the maths of oscillation — it slots directly into expressions like sin(ωt) and the formulas for simple harmonic motion, AC circuits, and waves — which is why physics and engineering favour ω over f even though they carry exactly the same information. A larger ω simply means a faster oscillation, in direct proportion to the frequency.
The formula is exact, but a couple of practical points are worth keeping in mind.
Frequency must be in hertz
This calculator expects the ordinary frequency f in hertz (cycles per second) and returns the angular frequency in radians per second. If your figure is in kilohertz or megahertz, convert it to hertz first — multiply kHz by 1000 or MHz by 1,000,000 — or the result will be off by that same factor. The conversion ω = 2πf assumes a steady, periodic oscillation; it does not apply to one-off events that never repeat.