Why is 206,265 such a special number in astronomy?

It's the number of arcseconds in one radian (180 × 3,600 ÷ π ≈ 206,265). A star at 1 parsec distance shows a parallax of exactly 1 arcsecond (= 1/206,265 radians). This constant defines the parsec and appears in virtually every telescope resolution and astrometric formula.

My telescope has a 200 mm aperture and I want to know the diffraction limit in arcseconds. How do I calculate?

θ (radians) = 1.22λ/D where λ is the wavelength. For λ = 550 nm (green): θ = 1.22 × 5.5×10⁻⁷ / 0.2 = 3.355×10⁻⁶ radians. Then: 3.355×10⁻⁶ × 206,265 ≈ 0.69 arcseconds — that's the diffraction-limited resolution.

A star's proper motion is 0.000004 radians per year. How many arcseconds per year is that?

0.000004 × 206,265 ≈ 0.825 arcseconds per year. Barnard's Star — the fastest-moving star in the sky — moves about 10.3 arcseconds per year, so 0.825 would be a fast-moving but not record-breaking proper motion.

My GPS software gives angular uncertainty in radians. I need to express it in arcseconds for a geodetic report.

Multiply by 206,265. For a typical GPS uncertainty of 3 × 10⁻⁵ radians: 3 × 10⁻⁵ × 206,265 ≈ 6.2 arcseconds of angular position uncertainty.

Radians to Arcseconds Converter

Enter a value to instantly convert between angle units.

From

Precision4 dp

1 Radian = 206264.8061 Arcsecond

Key Formulas

Degree → Radian

rad = ° × 0.0174533

Radian → Degree

° = rad × 57.2958

Degree → Gradian

grad = ° × 1.11111

Gradian → Degree

° = grad × 0.9

Formula

arcseconds = radians × 206264.806

The number 206,265 arcseconds per radian is one of the most important constants in observational astronomy and precision optics. It appears in the formula for stellar parallax (where 1 parsec = the distance at which 1 AU subtends 1 arcsecond), in the resolution formula for telescopes (θ = 1.22λ/D in radians, then multiplied by 206,265 to get arcseconds), and in astrometric plate-scale calculations. Whenever a physics or optics formula produces an angle in radians that must be reported in arcseconds, this constant is the bridge.

Source: ISO 80000-3:2019 (Quantities and units — Space and time)

Last reviewed: 2026-06-11 · see our methodology

Frequently Asked Questions

Real-World Examples

One radian — 206,265 arcseconds (≈ 57.3°). The fundamental astronomical conversion constant, defining the parsec and appearing in every telescope resolution formula.

1 rad = 206265 ″

A 1 mrad (milliradian) angle — 206.3 arcseconds. Used when comparing military mrad-based targeting angles with astronomical arcsecond measurements.

0.001 rad = 206.3 ″

One arcsecond in radians — 4.848 microradians, the angular resolution of the human eye and the limit of sub-arcsecond astrometry.

0.000004848 rad = 1 ″

The gravitational deflection of light by the Sun is 1.75 arcseconds = 8.49 × 10⁻⁶ radians — the value predicted by general relativity and confirmed by eclipse observations.