Convert Radians to Arcseconds - Angle Converter
This converter transforms an angle measured in radians into arcseconds, a common unit in astronomy, surveying, optics, and high-precision engineering. The conversion follows the SI-derived relationship between radians and degrees, and the definition of an arcsecond as 1/3600 of a degree.
Use this tool for quick lookups, verification of calculation pipelines, or preparing values for instrumentation (telescopes, theodolites, interferometers) where angular resolution is expressed in arcseconds. For traceable unit definitions and recommended notation see international standards and national metrology guidance.
Interactive Converter
Convert between radian and arcsecond with precision rounding.
Quick reference table
| Radian | Arcsecond |
|---|---|
| 1 rad | 206,264.806247 arcsec |
| 5 rad | 1,031,324.031235 arcsec |
| 10 rad | 2,062,648.062471 arcsec |
| 25 rad | 5,156,620.156177 arcsec |
| 50 rad | 10,313,240.312355 arcsec |
| 100 rad | 20,626,480.62471 arcsec |
Methodology
A radian is defined as the ratio of arc length to radius; it is the SI coherent unit for plane angle and is dimensionless. A full circle equals 2π radians.
An arcsecond is a subdivision of the degree: 1 degree = 60 minutes = 3600 arcseconds. Conversion uses the exact mathematical relationships between radians and degrees, and between degrees and arcseconds.
This converter applies the exact formula arcseconds = radians × (180 / π) × 3600. For convenience a high-precision constant (≈ 206264.80624709636 arcsec per radian) is used when displaying numeric outputs.
Worked examples
Example 1 — 1 radian: 1 rad × (180/π) × 3600 ≈ 206264.806247 arcseconds.
Example 2 — 1 degree in radians (≈ 0.017453292519943295 rad): 0.017453292519943295 rad × (180/π) × 3600 = 3600 arcseconds.
Example 3 — Convert a small angle commonly used in astrometry: 4.84813681109536e-6 rad ≈ 1 arcsecond (since 1 arcsecond = π / 648000 radians).
Further resources
External guidance
Expert Q&A
What is the exact mathematical relationship between radians and arcseconds?
arcseconds = radians × (180 / π) × 3600. Equivalently, 1 radian ≈ 206264.80624709636 arcseconds.
How many significant figures should I keep for instrument work?
Keep precision consistent with your instrument's resolution and uncertainty budget. For human-eye or handheld surveying, 1–2 significant figures in arcseconds is sufficient; for telescopes or interferometers, report to the resolution limit (sub-arcsecond) and include measurement uncertainty per ISO/IEC and NIST guidance.
How do I convert back from arcseconds to radians?
Use radians = arcseconds × (π / 180) / 3600, or radians = arcseconds × (π / 648000).
Why use radians instead of degrees when calculating?
Radians are the SI-coherent unit for angle and simplify many mathematical expressions (calculus, small-angle approximations). Convert to arcseconds only for reporting or comparing with instrumentation specifications.
Are there standard references for these unit definitions?
Yes. The SI brochure and national metrology institutes document unit definitions and recommended practices. Use those references for traceability and when preparing calibration or uncertainty statements.
What rounding or formatting is appropriate for automated pipelines?
Round results to a number of decimal places that preserves the meaningful digits given upstream uncertainty. For automated pipelines, preserve full double precision internally and round only at the final presentation layer; log conversion formulas and constants for reproducibility.
Sources & citations
- BIPM — The International System of Units (SI Brochure) — https://www.bipm.org/en/publications/si-brochure
- NIST — Guide to the SI (National Institute of Standards & Technology) — https://www.nist.gov/pml/weights-and-measures/si-units
- MIT OpenCourseWare — Calculus (definitions and angle measures) — https://ocw.mit.edu