What this calculator does
Angular frequency measures how fast something oscillates or rotates, expressed in radians per second rather than cycles per second. This calculator applies the standard angular frequency formula, omega equals 2 pi times frequency, to convert between frequency in hertz, period in seconds and angular frequency itself, starting from whichever one you already know.
The extra factor of 2 pi compared with ordinary frequency exists because one full cycle covers 2 pi radians, not one. A wave or rotation that completes one cycle per second, a frequency of 1 Hz, is turning through 2 pi radians every second, so its angular frequency is 2 pi radians per second, not 1.
The formula
Angular frequency equals 2 pi multiplied by frequency in hertz, or equivalently 2 pi divided by the period in seconds. Given any one of frequency, period or angular frequency, the other two follow directly from those same relationships.
| Term | Meaning |
|---|---|
| Angular frequency (ω) | The rate of oscillation in radians per second: 2π × frequency. |
| Frequency (f) | The number of complete cycles per second, measured in hertz. |
| Period (T) | The time taken for one complete cycle, in seconds: the reciprocal of frequency. |
The inputs explained
| Field | What to enter |
|---|---|
| Given value | Which quantity you already know, so the calculator can work out the other two. |
| Value | The value of that known quantity, in hertz, radians per second, or seconds depending on your selection above. |
When to use it
Working with simple harmonic motion
Formulas for oscillators, pendulums and springs are usually written in terms of angular frequency directly, so converting a measured frequency in hertz into radians per second is a common first step before using them.
Analysing AC electrical circuits
Reactance and impedance formulas for capacitors and inductors use angular frequency rather than plain frequency, so a mains frequency of 50 or 60 Hz needs converting before it can be substituted into those equations.
Converting a known period into a rate
When a rotation or cycle time is measured directly in seconds, this calculator converts that period straight into both frequency and angular frequency without an extra manual step.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How angular frequency compares to frequency across common values
A range of frequencies in hertz, converted to angular frequency.
| Frequency | Angular frequency | Period |
|---|---|---|
| 1 Hz | 6.283 rad/s | 1.0000 s |
| 10 Hz | 62.832 rad/s | 0.100000 s |
| 50 Hz | 314.159 rad/s | 0.020000 s |
| 60 Hz | 376.991 rad/s | 0.016667 s |
| 100 Hz | 628.319 rad/s | 0.010000 s |
| 440 Hz | 2,764.60 rad/s | 0.002273 s |
Questions
What is the angular frequency formula?
Angular frequency omega equals 2 pi multiplied by frequency f, written ω = 2πf. Since frequency is also the reciprocal of the period T, this is equally written as ω = 2π/T.
Why use angular frequency instead of ordinary frequency?
Many physics and engineering formulas, particularly for oscillation, waves and AC circuits, come out simpler when expressed in radians per second rather than cycles per second, because rotation and phase are naturally measured in radians.
Is angular frequency the same as angular velocity?
They share the same units and the same underlying idea of a rate measured in radians per second, but angular velocity usually describes an object physically rotating, while angular frequency describes the rate of a repeating cycle such as a wave or an oscillation.
What is the angular frequency of standard mains electricity?
For a 50 Hz supply it is 2π × 50, about 314.16 rad/s. For a 60 Hz supply it is 2π × 60, about 376.99 rad/s.
For the wavelength and speed relationships of a full wave, including its own angular frequency figure, see the wave speed, frequency and wavelength calculator. For rotational speed measured directly from angle and time, see the angular velocity calculator.