Minute and second of arc

  (Redirected from Milliarcsecond)

A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol ,[1] is a unit of angular measurement equal to 1/60 of one degree.[2] Since one degree is 1/360 of a turn (or complete rotation), one minute of arc is 1/21600 of a turn. The nautical mile (nmi) was originally defined as a minute of latitude on a spherical Earth, so the actual Earth circumference is very near 21600 nmi. A minute of arc is π/10800 of a radian.

Arcminute and football.png
An illustration of the size of an arcminute (not to scale). A standard association football (soccer) ball (with a diameter of 22 cm or 8.7 in) subtends an angle of 1 arcminute at a distance of approximately 756 m (827 yd).
General information
Unit systemNon-SI units mentioned in the SI
Unit ofAngle
Symbol or arcmin
In unitsDimensionless with an arc length of approx. ≈ 0.2909/1000 of the radius, i.e. 0.2909 mm/m
   degrees   1/60° = 0.016°
   arcseconds   60″
   radians   π/10800 ≈ 0.000290888 rad
   milliradians   π·1000/10800 ≈ 0.2909 mrad
   gons   9/600g = 0.015g
   turns   1/21600

A second of arc, arcsecond (arcsec), or arc second, denoted by the symbol ,[3] is 1/60 of an arcminute, 1/3600 of a degree,[2] 1/1296000 of a turn, and π/648000 (about 1/206181.8) of a radian.


To express even smaller angles, standard SI prefixes can be employed; the milliarcsecond (mas) and microarcsecond (μas), for instance, are commonly used in astronomy. For a three-dimensional area such as on a sphere, square arcminutes or seconds may be used.

Symbols and abbreviationsEdit

The prime symbol (U+2032) designates the arcminute,[1][3] though a single quote ' (U+0027) is commonly used where only ASCII characters are permitted. One arcminute is thus written as 1′. It is also abbreviated as arcmin or amin.

Similarly, double prime (U+2033) designates the arcsecond,[1][3] though a double quote " (U+0022) is commonly used where only ASCII characters are permitted. One arcsecond is thus written as 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, 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′.[4][5] This notation has been carried over into marine GPS receivers, which normally display latitude and longitude in the latter format by default.[6]

Common examplesEdit

The full moon's average apparent size is about 31 arcminutes (or 0.52°).

An arcminute is approximately the resolution of the human eye.

An arcsecond is approximately the angle subtended by a U.S. dime coin (18 mm) at a distance of 4 kilometres (about 2.5 mi).[7] An arcsecond is also the angle subtended by

  • an object of diameter one astronomical unit (149597870.7 km) at a distance of one parsec, per the definition of the latter.[8]

A milliarcsecond is about the size of a dime atop the Eiffel Tower, as seen from New York City.



Also notable examples of size in arcseconds are:


The concepts of degrees, minutes, and seconds—as they relate to the measure of both angles and time—derive from Babylonian astronomy and time-keeping. Influenced by the Sumerians, the ancient Babylonians divided the Sun's perceived motion across the sky over the course of one full day into 360 degrees.[10] Each degree was subdivided into 60 minutes and each minute into 60 seconds.[11][12] Thus, one Babylonian degree was equal to four minutes in modern terminology, one Babylonian minute to four modern seconds, and one Babylonian second to 1/15 (approximately 0.067) of a modern second.



Comparison of angular diameter of the Sun, Moon, planets and the International Space Station. True represent­ation of the sizes is achieved when the image is viewed at a distance of 103 times the width of the "Moon: max." circle. For example, if the "Moon: max." circle is 10 cm wide on a computer display, viewing it from 10.3 m (11.3 yards) away will show true representation of the sizes.

Since antiquity, the arcminute and arcsecond have been used in astronomy: in the ecliptic coordinate system as latitude (β) and longitude (λ); in the horizon system as altitude (Alt) and azimuth (Az); and in the equatorial coordinate system as declination (δ). All are measured in degrees, arcminutes, and arcseconds. The principal exception is right ascension (RA) in equatorial coordinates, which is measured in time units of hours, minutes, and seconds.

The arcsecond is also often used to describe small astronomical angles such as the angular diameters of planets (e.g. the angular diameter of Venus which varies between 10″ and 60″); the proper motion of stars; the separation of components of binary star systems; and parallax, the small change of position of a star or solar system body as the Earth revolves about the Sun. These small angles may also be written in milliarcseconds (mas), or thousandths of an arcsecond. The unit of distance called the parsec, abbreviated from the parallax angle of one arc second, was developed for such parallax measurements. The distance from the Sun to a celestial object is the reciprocal of the angle, measured in arcseconds, of the object's apparent movement caused by parallax.

The European Space Agency's astrometric satellite Gaia, launched in 2013, can approximate star positions to 7 microarcseconds (µas).[13]

Apart from the Sun, the star with the largest angular diameter from Earth is R Doradus, a red giant with a diameter of 0.05″.[note 1] Because of the effects of atmospheric blurring, ground-based telescopes will smear the image of a star to an angular diameter of about 0.5″; in poor conditions this increases to 1.5″ or even more. The dwarf planet Pluto has proven difficult to resolve because its angular diameter is about 0.1″.[14]

Space telescopes are not affected by the Earth's atmosphere but are diffraction limited. For example, the Hubble Space Telescope can reach an angular size of stars down to about 0.1″. Techniques exist for improving seeing on the ground. Adaptive optics, for example, can produce images around 0.05″ on a 10 m class telescope.


Minutes (′) and seconds (″) of arc are also used in cartography and navigation. At sea level one minute of arc along the equator equals exactly one geographical mile along the Earth's equator or approximately one nautical mile (1,852 metres; 1.151 miles).[15] A second of arc, one sixtieth of this amount, is roughly 30 metres (98 feet). The exact distance varies along meridian arcs or any other great circle arcs because the figure of the Earth is slightly oblate (bulges a third of a percent at the equator).

Positions are traditionally given using degrees, minutes, and seconds of arcs for latitude, the arc north or south of the equator, and for longitude, the arc east or west of the Prime Meridian. Any position on or above the Earth's reference ellipsoid can be precisely given with this method. However, when it is inconvenient to use base-60 for minutes and seconds, positions are frequently expressed as decimal fractional degrees to an equal amount of precision. Degrees given to three decimal places (1/1000 of a degree) have about 1/4 the precision of degrees-minutes-seconds (1/3600 of a degree) and specify locations within about 120 metres (390 feet). For navigational purposes positions are given in degrees and decimal minutes, for instance The Needles lighthouse is at 50º 39.734’N 001º 35.500’W.[16]

Property cadastral surveyingEdit



Example ballistic table for a given 7.62×51mm NATO load. Bullet drop and wind drift are shown both in mrad and minute of angle.

The arcminute is commonly found in the firearms industry and literature, particularly concerning the precision of rifles, though the industry refers to it as minute of angle (MOA). It is especially popular as a unit of measurement with shooters familiar with the imperial measurement system because 1 MOA subtends a circle with a diameter of 1.047 inches (which is often rounded to just 1 inch) at 100 yards (2.66 cm at 91 m or 2.908 cm at 100 m), a traditional distance on American target ranges. The subtension is linear with the distance, for example, at 500 yards, 1 MOA subtends 5.235 inches, and at 1000 yards 1 MOA subtends 10.47 inches. Since many modern telescopic sights are adjustable in half (1/2), quarter (1/4) or eighth (1/8) MOA increments, also known as clicks, zeroing and adjustments are made by counting 2, 4 and 8 clicks per MOA respectively.



Another common system of measurement in firearm scopes is the milliradian (mrad). Zeroing an mrad based scope is easy for users familiar with base ten systems. The most common adjustment value in mrad based scopes is 1/10 mrad (which approximates 13 MOA).

  • To adjust a 1/10 mrad scope 0.9 mrad down and 0.4 mrad right, the scope needs to be adjusted 9 clicks down and 4 clicks right (which equals approximately 3 and 1.5 MOA respectively).

One thing to be aware of is that some MOA scopes, including some higher-end models,[citation needed] are calibrated such that an adjustment of 1 MOA on the scope knobs corresponds to exactly 1 inch of impact adjustment on a target at 100 yards, rather than the mathematically correct 1.047 inches. This is commonly known as the Shooter's MOA (SMOA) or Inches Per Hundred Yards (IPHY). While the difference between one true MOA and one SMOA is less than half of an inch even at 1000 yards,[17] this error compounds significantly on longer range shots that may require adjustment upwards of 20–30 MOA to compensate for the bullet drop. If a shot requires an adjustment of 20 MOA or more, the difference between true MOA and SMOA will add up to 1 inch or more. In competitive target shooting, this might mean the difference between a hit and a miss.


Sometimes, a precision-oriented firearm's performance will be measured in MOA. This simply means that under ideal conditions (i.e. no wind, high-grade ammo, clean barrel, and a stable mounting platform such as a vise or a benchrest used to eliminate shooter error), the gun is capable of producing a group of shots whose center points (center-to-center) fit into a circle, the average diameter of circles in several groups can be subtended by that amount of arc. For example, a 1 MOA rifle should be capable, under ideal conditions, of repeatably shooting 1-inch groups at 100 yards. Most higher-end rifles are warrantied by their manufacturer to shoot under a given MOA threshold (typically 1 MOA or better) with specific ammunition and no error on the shooter's part. For example, Remington's M24 Sniper Weapon System is required to shoot 0.8 MOA or better, or be rejected from sale by quality control.

Rifle manufacturers and gun magazines often refer to this capability as sub-MOA, meaning a gun consistently shooting groups under 1 MOA. This means that a single group of 3 to 5 shots at 100 yards, or the average of several groups, will measure less than 1 MOA between the two furthest shots in the group, i.e. all shots fall within 1 MOA. If larger samples are taken (i.e., more shots per group) then group size typically increases, however this will ultimately average out. If a rifle was truly a 1 MOA rifle, it would be just as likely that two consecutive shots land exactly on top of each other as that they land 1 MOA apart. For 5-shot groups, based on 95% confidence, a rifle that normally shoots 1 MOA can be expected to shoot groups between 0.58 MOA and 1.47 MOA, although the majority of these groups will be under 1 MOA. What this means in practice is if a rifle that shoots 1-inch groups on average at 100 yards shoots a group measuring 0.7 inches followed by a group that is 1.3 inches, this is not statistically abnormal.[18][19]

The metric system counterpart of the MOA is the milliradian (mrad or 'mil'), being equal to one 1000th of the target range, laid out on a circle that has the observer as centre and the target range as radius. The number of milliradians on a full such circle therefore always is equal to 2 × π × 1000, regardless the target range. Therefore, 1 MOA ≈ 0.2909 mrad. This means that an object which spans 1 mrad on the reticle is at a range that is in metres equal to the object's size in millimetres[dubious ] (e.g. an object of 100 mm subtending 1 mrad is 100 metres away). So there is no conversion factor required, contrary to the MOA system. A reticle with markings (hashes or dots) spaced with a one mrad apart (or a fraction of a mrad) are collectively called a mrad reticle. If the markings are round they are called mil-dots.

In the table below conversions from mrad to metric values are exact (e.g. 0.1 mrad equals exactly 10 mm at 100 metres), while conversions of minutes of arc to both metric and imperial values are approximate.

Comparison of milliradian (mrad) and minute of arc (MOA).
Conversion of various sight adjustment increment
or click
of arc
At 100 m At 100 yd
(mm) (cm) (in) (in)
112 0.083′ 0.024 mrad 2.42 mm 0.242 cm 0.0958 in 0.087 in
0.2510 mrad 0.086′ 0.025 mrad 2.5 mm 0.25 cm 0.0985 in 0.09 in
18 0.125′ 0.036 mrad 3.64 mm 0.36 cm 0.144 in 0.131 in
16 0.167′ 0.0485 mrad 4.85 mm 0.485 cm 0.192 in 0.175 in
0.510 mrad 0.172′ 0.05 mrad 5 mm 0.5 cm 0.197 in 0.18 in
14 0.25′ 0.073 mrad 7.27 mm 0.73 cm 0.29 in 0.26 in
110 mrad 0.344′ 0.1 mrad 10 mm 1 cm 0.39 in 0.36 in
12 0.5′ 0.145 mrad 14.54 mm 1.45 cm 0.57 in 0.52 in
1.510 mrad 0.516′ 0.15 mrad 15 mm 1.5 cm 0.59 in 0.54 in
210 mrad 0.688′ 0.2 mrad 20 mm 2 cm 0.79 in 0.72 in
1′ 1.0′ 0.291 mrad 29.1 mm 2.91 cm 1.15 in 1.047 in
1 mrad 3.438′ 1 mrad 100 mm 10 cm 3.9 in 3.6 in
  • 1′ at 100 yards is about 1.047 inches[20]
  • 1′ ≈ 0.291 mrad (or 29.1 mm at 100 m, approximately 30 mm at 100 m)
  • 1 mrad ≈ 3.44′, so 1/10 mrad ≈ 1/3
  • 0.1 mrad equals exactly 1 cm at 100 m, or approximately 0.36 inches at 100 yards

Human visionEdit





Some measurement devices make use of arcminutes and arcseconds to measure angles when the object being measured is too small for direct visual inspection. For instance, a toolmaker's optical comparator will often include an option to measure in "minutes and seconds".

See alsoEdit


  1. ^ Some studies have shown a larger angular diameter for Betelgeuse. Various studies have produced figures of between 0.042 and 0.069″ for the star's diameter. The variability of Betelgeuse and difficulties in producing a precise reading for its angular diameter make any definitive figure conjectural.


  1. ^ a b c "List of Geometry and Trigonometry Symbols". Math Vault. 17 April 2020. Retrieved 31 August 2020.
  2. ^ a b Weisstein, Eric W. "Arc Second". mathworld.wolfram.com. Retrieved 31 August 2020.
  3. ^ a b c "Minutes of Arc to Degree Conversion". Inch Calculator. Retrieved 25 July 2021.
  8. ^ "Cosmic Distance Scales - The Milky Way".
  9. ^ a b "The Diffraction Limit of a Telescope".
  10. ^ "Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?". Scientific American. SCIENTIFIC AMERICAN, a Division of Springer Nature America, Inc. 5 March 2008. Retrieved 25 July 2021.
  11. ^ Correll, Malcolm (November 1977). "Early Time Measurements". The Physics Teacher. 15 (8): 476–479. doi:10.1119/1.2339739.
  12. ^ F. Richard Stephenson; Louay J. Fatoohi (May 1994). "The Babylonian Unit of Time". Journal for the History of Astronomy. doi:10.1177/002182869402500203.
  13. ^ Amos, Jonathan (14 September 2016). "Celestial mapper plots a billion stars". BBC News. Retrieved 31 March 2018.
  14. ^ NASA.gov Pluto Fact Sheet
  15. ^ Kaplan, George H. (1 January 2003). "Nautical mile approximates an arcminute". Ocean Navigator. Navigator Publishing. Retrieved 22 March 2017.
  16. ^ The Corporation of Trinity House (10 January 2020). "1/2020 Needles Lighthouse". Notices to Mariners. Retrieved 24 May 2020.
  17. ^ Mann, Richard (18 February 2011). "Mil, MOA or inches?". Shooting Illustrated. Archived from the original on 10 November 2013. Retrieved 13 April 2015.
  19. ^ Bramwell, Denton (January 2009). "Group Therapy The Problem: How accurate is your rifle?". Varmint Hunter. 69. Archived from the original on 7 October 2011. Retrieved 21 May 2009.
  20. ^ Dexadine Ballistics Software – ballistic data for shooting and reloading. See Talk

External linksEdit