Convert Hertz to Radians per Second - Frequency Converter
This converter transforms ordinary frequency (cycles per second, measured in hertz) into angular frequency (radians per second), commonly used in physics and engineering for oscillatory systems, control, and signal processing.
The conversion follows SI conventions and is exact: angular frequency ω (rad/s) equals 2π times frequency f (Hz). Use this tool for quick checks, engineering calculations, and instrument readouts; see methodology and FAQs for guidance on measurement limits and calibration.
For traceability and recommended practice when reporting measurements, follow national and international SI guidance and calibrate instruments against accredited standards where high accuracy is required.
Governance
Record 84f163c51ebb • Reviewed by Fidamen Standards Committee
Interactive Converter
Convert between hertz and radian per second with precision rounding.
Quick reference table
| Hertz | Radian per Second |
|---|---|
| 1 Hz | RAD 0.00 rad/s |
| 5 Hz | RAD 0.00 rad/s |
| 10 Hz | RAD 0.00 rad/s |
| 25 Hz | RAD 0.00 rad/s |
| 50 Hz | RAD 0.00 rad/s |
| 100 Hz | RAD 0.00 rad/s |
Methodology
The conversion uses the fixed mathematical relationship ω = 2π·f. Here f is frequency in hertz (cycles per second) and ω is angular frequency in radians per second. The factor 2π converts cycles to radians because one cycle equals 2π radians.
This relationship is dimensionally consistent within the SI system: hertz is s⁻¹ and radians are dimensionless, so rad/s is equivalent to s⁻¹ scaled by 2π. Numerical results are deterministic; any observed uncertainty comes from the measurement of f, not the conversion itself.
When using measured frequency values, include instrument uncertainty, sampling limits (Nyquist), and bandwidth. For traceable measurements, use calibrated instruments and document uncertainty according to NIST/BIPM guidance.
Worked examples
1 Hz → 2π × 1 = 6.283185307179586 rad/s (approx. 6.28319 rad/s).
60 Hz → 2π × 60 = 376.99111843077515 rad/s (approx. 376.99112 rad/s).
0.5 Hz → 2π × 0.5 = 3.141592653589793 rad/s (π rad/s).
F.A.Q.
What is the exact mathematical relationship between Hz and rad/s?
Angular frequency ω in rad/s equals 2π times frequency f in Hz: ω = 2π·f. The conversion factor 2π is exact because one cycle equals 2π radians.
Do I need to worry about units or dimensions when converting?
No additional unit conversion is required beyond multiplying by 2π. Hertz is s⁻¹ and radians are dimensionless in SI, so the conversion is dimensionally consistent. Any uncertainty arises from how f was measured, not the arithmetic conversion.
How many significant figures should I report?
Report the converted value to no more significant figures than are justified by the original frequency measurement and the instrument's uncertainty. Include uncertainty estimates when precision matters, following guidance on measurement uncertainty.
How does instrument calibration affect the converted result?
Calibration determines the accuracy of the measured frequency f. Since the conversion is exact, errors in ω stem from errors in f. Use calibrated instruments traceable to national standards and follow documented uncertainty budgets for high-stakes measurements.
Are there practical limits I should know about (sampling, Nyquist, bandwidth)?
Yes. If f is derived from sampled data, ensure the sample rate exceeds twice the highest frequency component (Nyquist) to avoid aliasing. Instrument bandwidth and sampling jitter also influence the effective uncertainty of f and therefore ω.
How do I convert back from rad/s to Hz?
Invert the relationship: f = ω / (2π). Divide the angular frequency in rad/s by 2π to get frequency in hertz.
Where is this relationship documented authoritatively?
The ω = 2π·f relationship is standard in SI-consistent treatments of frequency and angular frequency. For authoritative references on SI units and measurement best practice, consult national metrology institutes and the SI brochure.
Sources & citations
- NIST — SI Units and Metrology Guidance — https://www.nist.gov/pml/si-units
- BIPM — The International System of Units (SI Brochure) — https://www.bipm.org/en/publications/si-brochure
- MIT OpenCourseWare — Vibrations and Waves (lecture material on angular frequency) — https://ocw.mit.edu/courses/8-03sc-physics-iii-vibrations-and-waves-fall-2016/
- NIST — Measurement Uncertainty and Traceability Guidance — https://www.nist.gov/pml/weights-and-measures/measurement-uncertainty
- ISO 80000-3:2019 — Space and time — https://www.iso.org/standard/64974.html
Further resources
Related tools
External guidance
Versioning & Change Control
Audit record (versions, QA runs, reviewer sign-off, and evidence).
Record ID: 84f163c51ebbWhat changed (latest)
v1.0.0 • 2025-11-10 • MINOR
Initial publication and governance baseline.
Why: Published with reviewed formulas, unit definitions, and UX controls.
Public QA status
PASS — golden 25 + edge 120
Last run: 2026-01-23 • Run: golden-edge-2026-01-23
Versioning & Change Control
Audit record (versions, QA runs, reviewer sign-off, and evidence).
What changed (latest)
v1.0.0 • 2025-11-10 • MINOR
Initial publication and governance baseline.
Why: Published with reviewed formulas, unit definitions, and UX controls.
Public QA status
PASS — golden 25 + edge 120
Last run: 2026-01-23 • Run: golden-edge-2026-01-23
Engine
v1.0.0
Data
Baseline (no external datasets)
Content
v1.0.0
UI
v1.0.0
Governance
Last updated: Nov 10, 2025
Reviewed by: Fidamen Standards Committee (Review board)
Credentials: Internal QA
Risk level: low
Reviewer profile (entity)
Fidamen Standards Committee
Review board
Internal QA
Entity ID: https://fidamen.com/reviewers/fidamen-standards-committee#person
Semantic versioning
- MAJOR: Calculation outputs can change for the same inputs (formula, rounding policy, assumptions).
- MINOR: New features or fields that do not change existing outputs for the same inputs.
- PATCH: Bug fixes, copy edits, or accessibility changes that do not change intended outputs except for previously incorrect cases.
Review protocol
- Verify formulas and unit definitions against primary standards or datasets.
- Run golden-case regression suite and edge-case suite.
- Record reviewer sign-off with credentials and scope.
- Document assumptions, limitations, and jurisdiction applicability.
Assumptions & limitations
- Uses exact unit definitions from the Fidamen conversion library.
- Internal calculations use double precision; display rounding follows the unit's configured decimal places.
- Not a substitute for calibrated instruments in regulated contexts.
- Jurisdiction-specific rules may require official guidance.
Change log
v1.0.0 • 2025-11-10 • MINOR
Initial publication and governance baseline.
Why: Published with reviewed formulas, unit definitions, and UX controls.
Areas: engine, content, ui • Reviewer: Fidamen Standards Committee • Entry ID: 2ad82dad3fa4