Minute and second of arc

Arcminute
An illustration of the size of an arcminute (not to scale.) A standard association football ball (22 cm diameter) subtends an angle of 1 arcminute at a distance of approximately 775 meters.
General information
Unit systemNon-SI units mentioned in the SI
Unit ofAngle
Symbol or arcmin
In unitsDimensionless with an arc length of approx. ≈ 0.2908/1000 of the radius, i.e. 0.2908 mm/m
Conversions
1 in ...... is equal to ...
degrees   1/60° = 0.016°
arcseconds   60′′
gons   600/9g = 66.6g
turns   1/21600

A minute of arc, arcminute (arcmin), arc minute, or minute arc is a unit of angular measurement equal to 1/60 of one degree. Since one degree is 1/360 of a turn (or complete rotation), one minute of arc is 1/21600 of a turn – it is for this reason that the Earth's circumference is almost exactly 21,600 nautical miles. A minute of arc is π/10800 of a radian.

A second of arc, arcsecond (arcsec), or arc second is 1/60 of an arcminute, 1/3600 of a degree, 1/1296000 of a turn, and π/648000 (about 1/206265) of a radian.These units originated in Babylonian astronomy as sexagesimal subdivisions of the degree; they are used in fields that involve very small angles, such as astronomy, optometry, ophthalmology, optics, navigation, land surveying, and marksmanship.

To express even smaller angles, standard SI prefixes can be employed; the milliarcsecond (mas) and microarcsecond (μas), for instance, are commonly used in astronomy.

The number of square arcminutes in a complete sphere is ${\displaystyle 4\pi \left({\frac {10\,800}{\pi }}\right)^{2}={\frac {466\,560\,000}{\pi }}\approx }$ 148510660 square arcminutes (the surface area of a unit sphere in square units divided by the solid angle area subtended by a square arcminute, also in square units – so that the final result is a dimensionless number).

Symbols and abbreviations

The standard[1] symbol for marking the arcminute is the prime (′) (U+2032), though a single quote (') (U+0027) is commonly used where only ASCII characters are permitted. One arcminute is thus written 1′. It is also abbreviated as arcmin or amin or, less commonly, the prime with a circumflex over it (${\displaystyle {\hat {'}}}$).

The standard[1] symbol for the arcsecond is the double prime (″) (U+2033), though a double quote (") (U+0022) is commonly used where only ASCII characters are permitted. One arcsecond is thus written 1″. It is also abbreviated as arcsec or asec.

Sexagesimal system of angular measurement
Unit Value Symbol Abbreviations In radians, approx.
Degree 1/360 turn ° (degree) deg 17.4532925 mrad
Arcminute 1/60 degree ′ (prime) arcmin, amin, am, ${\displaystyle {\hat {'}}}$, MOA 290.8882087 μrad
Arcsecond 1/60 arcminute = 1/3600 degree ″ (double prime) arcsec, asec, as 4.8481368 μrad
Milliarcsecond 0.001 arcsecond = 1/3600000 degree mas 4.8481368 nrad
Microarcsecond 0.001 mas = 0.000001 arcsecond μas 4.8481368 prad

In celestial navigation, seconds of arc are rarely used in calculations, the preference usually being for degrees, minutes and decimals of a minute, for example, written as 42° 25.32′ or 42° 25.322′.[2][3] This notation has been carried over into marine GPS receivers, which normally display latitude and longitude in the latter format by default.[4]