Hz to Radians Converter — Hz ↔ rad/s, RPM, deg/s Calculator | nrtechstudio

Angular frequency lab · ω = 2πf

Hz to Radians Converter

Convert hertz to radians per second and back — this hz to rads calculator (a hz to radians calculator) handles hz to rads conversion and any hz to radians per second lookup: type hertz (hz) to get radians per second (rad/s) — also called the hz to radian per second converter — plus RPM, revolutions per second, degrees per second, period, motor shaft speed, tip speed, and oscilloscope waveform angular frequency. One formula, every angular unit, zero guesswork.

Angular velocity ω

314.1593 rad/s

= 50 Hz · 3000 RPM · T = 20 ms

100% client-side — nothing you type ever leaves your browser

01 · ConverterHz ↔ rad/s, two-way

Enter a value in any angular or frequency unit — convert hz to rev/s, rev/s to rad/s, deg/s to hz, hz to degrees per second (hz to deg/s), or any pair; every equivalent updates live. This doubles as a frequency to period calculator for any hz to period lookup, converting hz to angular velocity and hz to angular frequency alike (angular frequency from hz uses the same ω = 2πf).

Value (non-negative)

ω = 2π × 50 Hz = 314.1593 rad/s

rad/s

314.1593

RPM

3000

rev/s

deg/s

18000

Period T

20 ms

Core relation — angular velocity from frequency: ω (rad/s) = 2π × f (Hz), because one full cycle spans 2π radians.

04 · GuideFormulas, examples & reference

Everything below is verified against an independent Python oracle (33 test vectors) before this page was built.

The core formulas

ω (rad/s) = 2π × f (Hz) · f (Hz) = ω (rad/s) / 2π

RPM = f × 60 · ω = RPM × 2π/60 · deg/s = f × 360 · T = 1/f

Worked examples

Example 1 — AC mains, 60 Hz → rad/s

ω = 2π × 60 = 376.9911 rad/s T = 1/60 = 0.016667 s = 16.6667 ms

Answer: 60 Hz = 376.9911 rad/s (a 2-pole synchronous motor spins at 3600 RPM here).

Example 2 — Motor nameplate, 1725 RPM → rad/s

f = 1725 / 60 = 28.75 Hz ω = 2π × 28.75 = 180.6416 rad/s

Answer: 1725 RPM = 180.6416 rad/s.

Example 3 — Oscilloscope, 440 Hz (concert A) → ω

ω = 2π × 440 = 2764.6015 rad/s T = 1/440 = 0.0022727 s = 2.2727 ms waveform: y = A·sin(2764.6015·t + φ)

Answer: 440 Hz = 2764.6015 rad/s, period 2.2727 ms.

1 Hz equals

Unit	Value
Radians per second	6.2831853 rad/s
Revolutions per minute	60 RPM
Revolutions per second	1 rev/s
Degrees per second	360 deg/s
Period	1 s

Hz → rad/s conversion table

Hertz (Hz)	rad/s	RPM	Context
1	6.2832	60	1 rev per second
2	12.5664	120	—
5	31.4159	300	—
10	62.8319	600	—
20	125.6637	1200	—
50	314.1593	3000	AC mains (EU/Asia)
60	376.9911	3600	AC mains (N. America)
100	628.3185	6000	—
200	1256.6371	12000	—
300	1884.9556	18000	—
440	2764.6015	26400	Musical A4
500	3141.5927	30000	—
1000	6283.1853	60000	1 kHz reference tone

How to read the table: 1 hz in rad/s = 6.2832 (1 hz to rad/s); 100 hz to rad/s = 628.3185; 300 hz to rad/s = 1884.9556; 1000 hz to rad/s = 6283.1853; 1725 rpm to rad/s = 180.6416; 3000 rpm to rad/s = 314.1593; 3600 rpm to rad/s = 376.9911; ac 50 hz to rad/s = 314.1593; ac 60 hz to rad/s = 376.9911 (mains frequency rad/s values).

Which tool do you need? — scope boundary

This page (hz-to-rads) is the angular specialist: Hz ↔ rad/s ↔ RPM ↔ rev/s ↔ deg/s, motor shaft speed, tip speed, and waveform angular frequency.

If you arrived from a frequency converter angular-units search, you are in the right place: hz to khz vs hz to rad/s — the first only rescales within one unit family, the second converts between unit families.

Frequency Converter hub (nrtechstudio.com/frequency-converter/) covers SI prefix scaling — Hz ↔ kHz ↔ MHz ↔ GHz ↔ THz — plus wavelength, radio bands, and Nyquist analysis. If your question is "how many MHz is 2.4 GHz", go there.

Rule of thumb: changing the size prefix → the hub; changing rotation/angle units → this page.

Methodology

All conversions are exact multiplications/divisions by the factor 2π (or 60, 360) using IEEE-754 double precision (~15–16 significant digits).
Every engine function was verified against an independent Python oracle on 33 frozen test vectors — including 50/60 Hz mains values, 1725/3000 RPM motors, a 440 Hz waveform, round-trips, and magnitude extremes (1 µHz, 2.4 GHz) — with zero mismatches before release.
Period T = 1/f is exact by definition; displayed values are rounded for readability while the underlying math keeps full precision.
Synchronous speed uses the standard induction-motor relation n_s = 120f/P (f in Hz, P poles).

Honest limitations

Frequency in Hz is a magnitude — this converter accepts non-negative values only. Signed angular velocity (rotation direction) is a physics convention, not a converter input.
Motor figures are kinematic: real induction motors run below synchronous speed (slip), and tip speed ignores vibration, balance, and material limits.
Results are for calculation, learning, and design estimates — verify independently before safety-critical engineering.

Frequently asked questions

What is the formula to convert Hz to rad/s?

The hz to rads formula is ω = 2πf: the angular frequency (or angular velocity) ω in radians per second equals 2π times the frequency f in hertz. Here is how to convert hz to rads (how to convert hz to rad/s) in one step — multiply the hertz value by 6.2831853. Since one full cycle (one revolution) sweeps exactly 2π radians, each hertz contributes 6.2831853 rad/s. Example: 60 Hz → ω = 2π × 60 ≈ 376.99 rad/s.

How do you convert rad/s back to Hz?

To convert rads to hz — radians per second to hz, or radian to hz in the singular — divide the angular value by 2π: f = ω / (2π). That is how to convert rad/s to hz in any phrasing. For example, 300 rad/s ÷ 6.2831853 ≈ 47.75 Hz. This tool converts both directions instantly — just pick the units above.

Is Hz the same as rad/s?

No — is hz the same as rad/s? They are not. Hertz counts whole cycles (or revolutions) per second, while radians per second measures angular change per second. The difference between hz and rad/s is exactly the factor 2π: 1 Hz = 6.2831853 rad/s. Treating them as equal is the single most common mistake in vibration and motor calculations.

Why is the factor 2π and not something else?

Because one complete cycle — one full turn of a shaft, one full oscillation of a sine wave — spans exactly 2π radians (360°). So a frequency of f cycles per second sweeps f × 2π radians per second. That is why you multiply by 2 pi when converting hz to rad/s: the 2π is pure geometry, not a convention. If you searched 'why 2 pi in hz to rad/s' or 'why multiply by 2 pi', this is the whole answer: one cycle is 2π radians.

Is angular frequency the same as angular velocity?

Numerically they share the same unit (rad/s) and the same formula ω = 2πf, but they describe different things: angular velocity is a vector describing how fast something physically rotates (direction included), while angular frequency is a scalar describing how fast an oscillation or wave repeats. The difference between angular frequency and angular velocity is conceptual, not numerical — for a steadily spinning shaft they are equal in magnitude.

What is 50 Hz in rad/s? What about 60 Hz?

50 hz to rad/s = 2π × 50 ≈ 314.16 rad/s (European/Asian mains power). 60 hz to rad/s = 2π × 60 ≈ 376.99 rad/s (North American mains). You can verify both with the converter above — they are also the exact angular speeds of a 2-pole synchronous motor at those grid frequencies.

How do I convert RPM to rad/s?

To convert rpm to rad/s, multiply RPM by 2π/60 (≈ 0.10472): ω = RPM × 0.10472. For example, 3000 rpm to rad/s = 3000 × 0.10472 ≈ 314.16 rad/s. Equivalently, divide RPM by 60 to get Hz, then multiply by 2π. Use the Motor tab above to do this with tip-speed calculation included.

How do I convert rad/s to RPM?

To convert rad/s to rpm, multiply by 60/(2π) (≈ 9.5493): RPM = ω × 9.5493. Example: 100 rad/s → 954.93 RPM. Physicists often quote motor speeds in rad/s while nameplates use RPM — this is the bridge between them.

How many degrees per second is 1 Hz?

1 hz in deg/s is exactly 360 degrees per second. One cycle equals one full 360° turn, so deg/s = Hz × 360. For example, 50 Hz = 18,000 deg/s. Radians are preferred in physics because calculus on trig functions is cleaner with radians, but degrees per second is still common in robotics and servo specs.

What is the period of a 50 Hz or 60 Hz signal?

Period T = 1/f. The period of 50 hz is 20 ms; the period of 60 hz is 16.67 ms. The converter shows the period for every value you enter, because oscilloscopes measure time (period) while nameplates quote frequency — T = 1/f is how you move between them.

My oscilloscope shows 440 Hz. What is the angular frequency?

440 hz to rad/s: ω = 2π × 440 ≈ 2764.60 rad/s, with a period of about 2.273 ms. This is the ω that appears in the sine-wave equation y = A·sin(ωt + φ). The Wave tab above plots the actual waveform for any frequency you enter.

Can frequency or angular velocity be negative?

Ordinary frequency in Hz is never negative — it counts cycles, so can hz be negative? No. Angular velocity, however, can carry a sign indicating direction of rotation (clockwise vs counter-clockwise); negative frequency in signal processing is a related convention for direction of rotation of a phasor. This converter works with magnitudes (non-negative values).

Why do physics formulas use ω (rad/s) instead of f (Hz)?

Because calculus is cleaner in radians: the derivative of sin(ωt) is ω·cos(ωt) only when ω is in rad/s — with f in Hz you would carry an extra 2π everywhere. That is why physics formulas use omega (ω) whenever a formula involves rotation angles, phases, or time derivatives of waves: ω is the natural variable. So why do physics formulas use omega? Because every rotation, phase, and wave derivative is simpler without a stray 2π.

What is the difference between this page and the Frequency Converter hub?

Looking for a frequency converter for angular units? This page (hz-to-rads) is the angular specialist: Hz ↔ rad/s ↔ RPM ↔ rev/s ↔ deg/s, motor shaft speed, tip speed, and waveform angular frequency. Our Frequency Converter hub covers SI prefix scaling — Hz ↔ kHz ↔ MHz ↔ GHz ↔ THz — plus wavelength, radio bands, and Nyquist analysis. Unlike hz to khz (which only changes the size prefix), hz to rad/s changes the unit family. Rule of thumb: changing the size prefix → the hub; changing rotation/angle units → this page.

How accurate is this converter?

Conversions use full double-precision arithmetic with the exact factor 2π, verified against an independent Python oracle on 33 test vectors (including 50/60 Hz mains, 1725/3000 RPM motors, and a 440 Hz waveform). Displayed values are rounded for readability; the underlying math keeps ~15 significant digits. Results are for calculation and learning — not for safety-critical engineering without independent verification.

Is my data sent anywhere when I use this tool?

No. This converter runs entirely in your browser — every calculation happens locally in JavaScript. Nothing you type is uploaded, stored, or sent to any server, and the page works offline once loaded.

Free tool by NRTechStudio

Hz to Radians Converter

01 · ConverterHz ↔ rad/s, two-way

02 · Motor labShaft speed → angular velocity

03 · Wave labFrequency → waveform ω

Angular frequency ω

Angular velocity ω

04 · GuideFormulas, examples & reference

The core formulas

Worked examples

Example 1 — AC mains, 60 Hz → rad/s

Example 2 — Motor nameplate, 1725 RPM → rad/s

Example 3 — Oscilloscope, 440 Hz (concert A) → ω

1 Hz equals

Hz → rad/s conversion table

Which tool do you need? — scope boundary

Methodology

Honest limitations

Frequently asked questions

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