Article — Degrees to Seconds (Arcseconds) Converter
The Degrees to Arcseconds Converter
An arcsecond equals one three-thousand-six-hundredth of a degree, so the degrees to arcseconds formula is arcsec = deg × 3,600. On Earth, one arcsecond of arc covers about 30.87 metres at the equator. Telescopes routinely resolve features below 1 arcsecond, and modern astrometry reaches into microarcseconds.
The arcsecond (symbol ″) is the smallest standard sub-unit of the degree. It sits below the arcminute, which is itself one-sixtieth of a degree, giving the full chain 1° = 60′ = 3,600″. The arcsecond and its even smaller cousins — milliarcseconds and microarcseconds — drive almost every quantitative claim in modern astronomy and high-precision surveying.
What is an arcsecond?
An arcsecond is the angle subtended by an object whose width equals one part in 206,265 of its distance. That ratio comes directly from the conversion 1 arcsec = π / 648,000 radians, since 360 degrees equals 2π radians. In plain terms, a 1 cm object at a distance of about 2 kilometres covers roughly 1 arcsecond.
The arcsecond inherits Babylonian base-60 arithmetic. Each degree is divided into 60 minuta prima (arcminutes), and each arcminute into 60 minuta secunda (arcseconds). The notation has not changed since Ptolemy. Modern astronomers writing a star's declination as +45°30′15″ are using exactly the same system Hipparchus used in the 2nd century BCE.
The famous astronomical constant 206,265 is just the number of arcseconds in a radian (180° × 3,600 / π). It appears everywhere in astronomy because converting from radians to arcseconds is the most common unit-change in the field.
The degrees to arcseconds formula
The conversion is a single multiplication by 3,600. The factor is exact, so there is no rounding error built into the unit. Reverse the conversion by dividing by 3,600.
deg → arcsec multiply by 3,600arcsec → deg divide by 3,6001° = 60′ = 3,600″1 rad ≈ 206,265 arcsecFor a DMS reading, take the degree-minute-second triplet and convert each piece. D°M′S″ in decimal degrees equals D + M/60 + S/3,600. To get the same angle in arcseconds, multiply the decimal degree result by 3,600 — or, equivalently, compute D × 3,600 + M × 60 + S.
Arcseconds in astronomy and parallax
Arcseconds are the working unit for almost every astronomical measurement. Star catalogues list positions in degrees, arcminutes, and arcseconds. Proper motions — the apparent drift of stars across the sky over decades — are measured in arcseconds per year for the fastest movers, milliarcseconds per year for typical stars.
The parsec, the standard unit of stellar distance, is defined by arcseconds. A star at one parsec away shows an annual parallax shift of exactly 1 arcsecond as Earth orbits the Sun. The closest star to us, Proxima Centauri, has a parallax of about 0.77 arcsec — corresponding to a distance of 1.3 parsecs or 4.2 light-years. The Gaia space telescope routinely measures parallaxes to better than 50 microarcseconds for millions of stars.
Arcseconds on Earth — geodesy and GPS
On Earth's surface, arcseconds are tied to physical distance through the planet's size. One arcsecond of latitude is about 30.87 metres along any meridian. One arcsecond of longitude is also 30.87 metres at the equator, shrinking to zero at the poles as meridians converge.
Modern surveying instruments deliver angle measurements to the arcsecond level or better. A robotic total station can resolve 0.5 arcseconds — about 0.24 mm at 100 m — which is the floor for property-line surveying and bridge construction. GPS receivers in differential mode reach centimetre-scale accuracy, equivalent to milliarcsecond-scale angles.
- 1 arcsec latitude ≈ 30.87 m along any meridian
- 1 arcsec longitude (equator) ≈ 30.87 m, dropping toward poles
- USGS topographic survey baseline accuracy 1–5 arcsec
- Total station (robotic) 0.5 arcsec angular resolution
- RTK GPS ~1–2 cm precision (sub-arcsecond)
- Gravity gradient surveys work in microarcsec deflection units
Telescope resolution in arcseconds
The angular resolution of a telescope is set by the Rayleigh criterion — the minimum separation at which two point sources can be distinguished. For visible light, the formula gives resolution ≈ 138 / aperture, with aperture in millimetres and resolution in arcseconds. A 100 mm amateur scope resolves to ~1.4 arcsec; a 200 mm scope to ~0.7 arcsec.
Atmospheric seeing usually limits ground-based scopes to about 1 arcsec regardless of aperture, which is why the largest gains in resolution come from going to space (Hubble, JWST) or using adaptive optics. The Event Horizon Telescope, a virtual instrument built from radio telescopes across continents, achieves about 20 microarcseconds — fine enough to image the shadow of the supermassive black hole at the centre of the galaxy M87.
If you want to see Jupiter's moons as disks rather than points, you need about 1 arcsec resolution — well within reach of a backyard 100 mm scope on a steady night. The disks themselves are only 1–2 arcsec across.
Milliarcseconds and microarcseconds
For modern astrometry, even the arcsecond is too coarse. A milliarcsecond (mas) is 1/1000 of an arcsecond. A microarcsecond (μas) is one millionth. The European Space Agency's Gaia mission measures stellar positions to about 24 μas for the brightest stars, dropping to ~500 μas at the survey limit.
To put microarcseconds in perspective: 1 μas equals the angle subtended by a human hair at a distance of about 4,000 km. At that scale, Gaia can map the three-dimensional structure of the Milky Way to thousands of light-years, including individual stars in the bulge and halo.
Ground-based optical telescopes almost never beat 1 arcsecond resolution without adaptive optics. Atmospheric turbulence smears point sources across roughly that scale, so a 300 mm scope and a 3,000 mm scope often look the same in seeing-limited conditions. Space-based instruments avoid the problem entirely.
A short history of degrees and arcseconds
The Babylonian-derived 360° system has stayed in continuous use since at least 1500 BCE. The division of the degree into arcminutes and arcseconds dates to Ptolemy (around 150 CE), who wrote of "first sixtieths" and "second sixtieths" — minuta prima and minuta secunda. The Latin terms became "minute" and "second" in European languages, and we kept the system intact.
Tycho Brahe pushed naked-eye observations to about 1 arcminute precision in the late 1500s, a record that stood until telescopes pushed observations into arcseconds in the 1700s. Friedrich Bessel made the first measurement of stellar parallax in 1838, at about 0.3 arcsec for 61 Cygni. Gaia, launched in 2013, now measures the same kind of parallax for over a billion stars at microarcsecond precision.
Common degrees to arcseconds mistakes
Most arcsecond errors are unit-precedence errors — confusing arcseconds with seconds of time, or with arcminutes. A few patterns recur:
- arcsec vs second of time — astronomical right ascension uses seconds of time (1 hour = 15° at the equator), not arcseconds
- multiplying by 60 instead of 3,600 — that gives arcminutes, not arcseconds
- letting arcsec exceed 59 in DMS — 40°30′75″ should normalise to 40°31′15″
- treating 1 mas as 1 arcsec — a 1,000× error common in cross-mission astrometry tables
- over-quoting precision — GPS at 5 m precision is ~0.16 arcsec; quoting positions to milliarcseconds is meaningless without instrument context