Physics · 5 min read
How to Convert Angular Resolution from Radians to Arcseconds
θ″ = 206265 θ — the astronomy unit for a tiny Rayleigh angle
Rayleigh’s formula returns radians. Star charts, seeing reports, and double-star catalogs use arcseconds. Multiply θ by 206265. This tutorial converts the 10 cm / 550 nm result 6.71×10⁻⁶ rad into 1.38″ with the Angular Resolution Calculator.
Written by the My Calculator Stack editorial team. About our methods
206265 arcseconds per radian is the full circle 360×3600 divided by 2π. The factor does not change the physics; it only relabels a small angle. This is not angular velocity in rad/s.
1.Why the calculator prints two units
θ = 1.22 λ / D is a ratio of lengths, so the SI answer is radians. A milliradian is already huge for telescopes: 1″ is only 4.85×10⁻⁶ rad. Arcseconds keep the readout on the same scale as seeing (often 1–2″) and Dawes catalogs. The Angular Resolution Calculator therefore reports radians and arcseconds together. The Rayleigh identity itself is How to Calculate Angular Resolution with the Rayleigh Criterion.
\[ 1^{\prime\prime}=1/3600^{\circ}=4.848\times 10^{-6}\,\mathrm{rad} \]
2.The factor 206265
A full turn is 2π radians and also 360 × 3600 = 1,296,000 arcseconds. Divide: 1,296,000 / (2π) ≈ 206264.8, taught as 206265″ per radian. So θ″ = 206265 θ with θ in radians. Going the other way, θ = θ″ / 206265. Arcminutes are θ″ / 60; degrees are θ″ / 3600. Do not feed arcseconds into 1.22 λ / D — convert to radians first, or leave that conversion to the calculator.
\[ \theta^{\prime\prime}=206265\,\theta,\quad \theta=\theta^{\prime\prime}/206265 \]
3.Worked conversion: 6.71×10⁻⁶ rad
The visible default is λ = 550 nm, D = 0.10 m, θ = 1.22 × 5.50×10⁻⁷ / 0.10 = 6.71×10⁻⁶ rad. Then θ″ = 6.71×10⁻⁶ × 206265 = 1.384″, which rounds to 1.38″ on a two-decimal astronomy readout.
A radio check on the same factor: 0.0102 rad × 206265 ≈ 2114″ ≈ 35.2′. Type 550 nm and 0.1 m into the Angular Resolution Calculator and read both output fields; they should match this pair. The 21 cm / 25 m contrast is How Wavelength Affects Telescope Angular Resolution.
\[ 6.71\times 10^{-6}\times 206265\approx 1.38^{\prime\prime} \]
4.Small-angle length on the sky
Once θ is in radians, a transverse size at distance r is s ≈ r θ. With θ in arcseconds, s ≈ r θ″ / 206265. That is how a 1.38″ split at 1 km is only about 6.7 mm. Do not confuse that static geometry with angular velocity ω = θ / t on the Angular Velocity Calculator — a different θ, a rate. Camera pixels and Airy diameters also want arcseconds or microradians, not a raw 10⁻⁶ with no unit.
\[ s\approx r\theta=r\,\theta^{\prime\prime}/206265 \]
5.More angular resolution guides
The Rayleigh walkthrough: How to Calculate Angular Resolution with the Rayleigh Criterion. Aperture at fixed λ: How Aperture Diameter Sets Diffraction-Limited Resolution. Dawes in arcseconds: How to Compare Rayleigh and Dawes Limits for Double Stars. Camera pupil: How to Estimate the Diffraction Limit of a Camera or Lens. Seeing in arcseconds: How Atmospheric Seeing Limits Real-World Angular Resolution.
Try it yourself
Open the related calculator and put these formulas to work.