Minute and second of arc

For the SI units of angle, seeradianandmilliradian.

"Arcsec" redirects here. For arcsecant function, seeInverse trigonometric functions.

Arcminute
An illustration of the size of an arcminute (not to scale). A standardassociation 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 system	Non-SI units mentioned in the SI
Unit of	Angle
Symbol	′, arcmin
In units	Dimensionlesswith anarc lengthof approx. ≈⁠0.2909/1000⁠of the radius, i.e. 0.2909⁠mm/m⁠
Conversions
1′in...	... is equal to...
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

Aminute of arc,arcminute(arcmin),arc minute,orminute arc,denoted by the symbol′,is a unit ofangularmeasurement equal to⁠1/60⁠of onedegree.Since one degree is⁠1/360⁠of aturn, or complete rotation,one arcminute is⁠1/21600⁠of a turn. Thenautical mile(nmi) was originally defined as thearc lengthof a minute of latitude on a spherical Earth, so the actual Earth circumference is very near21600nmi.A minute of arc is⁠π/10800⁠of aradian.

Asecond of arc,arcsecond(arcsec), orarc second,denoted by the symbol″,is⁠1/60⁠of an arcminute,⁠1/3600⁠of a degree,⁠1/1296000⁠of a turn, and⁠π/648000⁠(about⁠1/206264.8⁠) of a radian.

These units originated inBabylonian astronomyassexagesimal(base 60) subdivisions of the degree; they are used in fields that involve very small angles, such asastronomy,optometry,ophthalmology,optics,navigation,land surveying,andmarksmanship.

To express even smaller angles, standardSI prefixescan be employed; themilliarcsecond(mas) andmicroarcsecond(μas), for instance, are commonly used in astronomy. For a three-dimensional area such as on a sphere,square arcminutesorsecondsmay be used.

Symbols and abbreviations

Theprime symbol′(U+2032) designates the arcminute,though a single quote'(U+0027) is commonly used where onlyASCIIcharacters are permitted. One arcminute is thus written as 1′. It is also abbreviated asarcminoramin.

Similarly,double prime″(U+2033) designates the arcsecond,though a double quote"(U+0022) is commonly used where onlyASCIIcharacters are permitted. One arcsecond is thus written as 1″. It is also abbreviated asarcsecorasec.

Sexagesimalsystem ofangular measurement

Unit	Value	Symbol		Abbreviations	In radians, approx.
Degree	⁠1/360⁠turn	°	Degree	deg	17.4532925mrad
Arcminute	⁠1/60⁠degree	′	Prime	arcmin, amin, am, MOA	290.8882087μrad
Arcsecond	⁠1/60⁠arcminute =⁠1/3600⁠degree	″	Double prime	arcsec, asec, as	4.8481368μrad
Milliarcsecond	0.001 arcsecond =⁠1/3600000⁠degree			mas	4.8481368nrad
Microarcsecond	0.001 mas =0.000001arcsecond			μas	4.8481368prad

Incelestial 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′.This notation has been carried over intomarine GPSand aviation GPS receivers, which normally display latitude and longitude in the latter format by default.

Common examples

The averageapparent diameterof thefull Moonis about 31 arcminutes, or 0.52°.

One arcminute is the approximate distance two contours can be separated by, and still be distinguished by, a person with20/20 vision.

One arcsecond is the approximateangle subtendedby aU.S. dime coin(18 mm) at a distance of 4 kilometres (about 2.5 mi).An arcsecond is also the angle subtended by

• an object of diameter725.27 kmat a distance of oneastronomical unit,
• an object of diameter45866916kmat onelight-year,
• an object of diameter one astronomical unit (149597870.7 km) at a distance of oneparsec,per the definition of the latter.

One milliarcsecond is about the size of a half dollar, seen from a distance equal to that between theWashington Monumentand theEiffel Tower.

One microarcsecond is about the size of a period at the end of a sentence in the Apollo mission manuals left on the Moon as seen from Earth.

One nanoarcsecond is about the size of a penny onNeptune's moonTritonas observed from Earth.

Also notable examples of size in arcseconds are:

• Hubble Space Telescopehas calculational resolution of 0.05 arcseconds and actual resolution of almost 0.1 arcseconds, which is close to thediffraction limit.
• At crescent phase,Venusmeasures between 60.2 and 66 seconds of arc.

History

The concepts of degrees, minutes, and seconds—as they relate to the measure of both angles and time—derive fromBabylonianastronomyand time-keeping. Influenced by theSumerians,the ancient Babylonians divided the Sun's perceived motion across the sky over the course of onefull dayinto 360 degrees.Each degree was subdivided into 60 minutes and each minute into 60 seconds.Thus, one Babylonian degree was equal to four minutes in modern terminology, one Babylonian minute to four modern seconds, and one Babylonian second to⁠1/15⁠(approximately 0.067) of a modern second.

Uses

Astronomy

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 inastronomy:in theecliptic coordinate systemas latitude (β) and longitude (λ); in thehorizon systemas altitude (Alt) andazimuth(Az); and in theequatorial coordinate systemasdeclination(δ). All are measured in degrees, arcminutes, and arcseconds. The principal exception isright ascension(RA) in equatorial coordinates, which is measured in time units of hours, minutes, and seconds.

Contrary to what one might assume, minutes and seconds of arc do not directly relate to minutes and seconds of time, in either the rotational frame of the Earth around its own axis (day), or the Earth's rotational frame around the Sun (year). The Earth's rotational rate around its own axis is 15 minutes of arc per minute of time (360 degrees / 24 hours in day); the Earth's rotational rate around the Sun (not entirely constant) is roughly 24 minutes of time per minute of arc (from 24 hours in day), which tracks the annual progression of the Zodiac. Both of these factor in what astronomical objects you can see from surface telescopes (time of year) and when you can best see them (time of day), but neither are in unit correspondence. For simplicity, the explanations given assume a degree/day in the Earth's annual rotation around the Sun, which is off by roughly 1%. The same ratios hold for seconds, due to the consistent factor of 60 on both sides.

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″); theproper motionof stars; the separation of components ofbinary star systems;andparallax,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 theparsec,abbreviated from theparallax angle of one arcsecond, was developed for such parallax measurements. The distance from the Sun to a celestial object is thereciprocalof the angle, measured in arcseconds, of the object's apparent movement caused by parallax.

TheEuropean Space Agency'sastrometricsatelliteGaia,launched in 2013, can approximate star positions to 7 microarcseconds (μas).

Apart from the Sun, the star with the largestangular diameterfrom Earth isR Doradus,ared giantwith a diameter of 0.05″. Because of the effects of atmosphericblurring,ground-basedtelescopeswill 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 planetPlutohas proven difficult to resolve because itsangular diameteris 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.

Space telescopes are not affected by the Earth's atmosphere but arediffraction limited.For example, theHubble Space Telescopecan reach an angular size of stars down to about 0.1″.

Cartography

Minutes (′) and seconds (″) of arc are also used incartographyandnavigation.Atsea levelone minute of arc along theequatorequals exactly onegeographical mile(not to be confused with international mile or statute mile) along the Earth's equator or approximately onenautical mile(1,852metres;1.151miles).A second of arc, one sixtieth of this amount, is roughly 30 metres (98 feet). The exact distance varies alongmeridian arcsor any othergreat circlearcs because thefigure of the Earthis slightlyoblate(bulges a third of a percent at the equator).

Positions are traditionally given using degrees, minutes, and seconds of arcs forlatitude,the arc north or south of the equator, and forlongitude,the arc east or west of thePrime Meridian.Any position on or above the Earth'sreference ellipsoidcan be precisely given with this method. However, when it is inconvenient to usebase-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 (⁠1/1000⁠of a degree) have about⁠1/4⁠the precision of degrees-minutes-seconds (⁠1/3600⁠of 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.

Property cadastral surveying

Related to cartography, property boundarysurveyingusing themetes and boundssystem andcadastral surveyingrelies on fractions of a degree to describe property lines' angles in reference tocardinal directions.A boundary "mete" is described with a beginning reference point, the cardinal direction North or South followed by an angle less than 90 degrees and a second cardinal direction, and a linear distance. The boundary runs the specified linear distance from the beginning point, the direction of the distance being determined by rotating the first cardinal direction the specified angle toward the second cardinal direction. For example,North 65° 39′ 18″ West 85.69 feetwould describe a line running from the starting point 85.69 feet in a direction 65° 39′ 18″ (or 65.655°) away from north toward the west.

Firearms

Example ballistic table for a given7.62×51mm NATOload. Bullet drop and wind drift are shown both inmradand minute of angle.

The arcminute is commonly found in thefirearmsindustry and literature, particularly concerning theprecisionofrifles,though the industry refers to it asminute of angle(MOA). It is especially popular as a unit of measurement with shooters familiar with theimperial measurement systembecause 1 MOAsubtendsa circle with a diameter of 1.047inches(which is often rounded to just 1 inch) at 100yards(2.66 cm at 91 m or 2.908 cm at 100 m), a traditional distance on Americantarget ranges.Thesubtensionis 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 moderntelescopic sightsare adjustable in half (⁠1/2⁠), quarter (⁠1/4⁠) or eighth (⁠1/8⁠) MOA increments, also known asclicks,zeroingand adjustments are made by counting 2, 4 and 8 clicks per MOA respectively.

For example, if the point of impact is 3 inches high and 1.5 inches left of the point of aim at 100 yards (which for instance could be measured by using aspotting scopewith a calibrated reticle, or a target delineated for such purposes), the scope needs to be adjusted 3 MOA down, and 1.5 MOA right. Such adjustments are trivial when the scope's adjustment dials have a MOA scale printed on them, and even figuring the right number of clicks is relatively easy on scopes thatclickin fractions of MOA. This makes zeroing and adjustments much easier:

• To adjust a1⁄2MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 × 2 = 6 clicks down and 1.5 x 2 = 3 clicks right
• To adjust a1⁄4MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 x 4 = 12 clicks down and 1.5 × 4 = 6 clicks right
• To adjust a1⁄8MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 x 8 = 24 clicks down and 1.5 × 8 = 12 clicks right

Another common system of measurement in firearm scopes is themilliradian(mrad). Zeroing an mrad based scope is easy for users familiar withbase tensystems. The most common adjustment value in mrad based scopes is⁠1/10⁠mrad (which approximates1⁄3MOA).

• To adjust a⁠1/10⁠mrad 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, 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,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.

The physical group size equivalent tomminutes of arc can be calculated as follows: group size = tan(⁠m/60⁠) × distance. In the example previously given, for 1 minute of arc, and substituting 3,600 inches for 100 yards, 3,600 tan(⁠1/60⁠) ≈ 1.047 inches. Inmetric units1 MOA at 100 metres ≈ 2.908 centimetres.

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 agroup of shotswhose 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, a1 MOA rifleshould 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'sM24 Sniper Weapon Systemis required to shoot 0.8 MOA or better, or be rejected from sale byquality control.

Rifle manufacturers and gun magazines often refer to this capability assub-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.

Themetric systemcounterpart of the MOA is themilliradian(mrad or 'mil'), being equal to1⁄1000of 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 thereticleis at a range that is in metres equal to the object's size in millimetres(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 calledmil-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 minute of arc (MOA) andmilliradian(mrad).

Conversion of various sight adjustment increment

Increment, or click	(mins of arc)	(milli- radians)	At 100m			At 100yd
			(mm)	(cm)	(in)	(in)
1⁄12′	0.083′	0.024 mrad	2.42 mm	0.242 cm	0.0958 in	0.087 in
0.25⁄10mrad	0.086′	0.025 mrad	2.5 mm	0.25 cm	0.0985 in	0.09 in
1⁄8′	0.125′	0.036 mrad	3.64 mm	0.36 cm	0.144 in	0.131 in
1⁄6′	0.167′	0.0485 mrad	4.85 mm	0.485 cm	0.192 in	0.175 in
0.5⁄10mrad	0.172′	0.05 mrad	5 mm	0.5 cm	0.197 in	0.18 in
1⁄4′	0.25′	0.073 mrad	7.27 mm	0.73 cm	0.29 in	0.26 in
1⁄10mrad	0.344′	0.1 mrad	10 mm	1 cm	0.39 in	0.36 in
1⁄2′	0.5′	0.145 mrad	14.54 mm	1.45 cm	0.57 in	0.52 in
1.5⁄10mrad	0.516′	0.15 mrad	15 mm	1.5 cm	0.59 in	0.54 in
2⁄10mrad	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
• 1′ ≈ 0.291 mrad (or 29.1 mm at 100 m, approximately 30 mm at 100 m)
• 1 mrad ≈ 3.44′, so⁠1/10⁠mrad ≈⁠1/3⁠′
• 0.1 mrad equals exactly 1 cm at 100 m, or exactly 0.36 inches at 100 yards

Human vision

In humans,20/20 visionis the ability to resolve aspatial patternseparated by avisual angleof one minute of arc, from a distance of twentyfeet. A 20/20 letter subtends 5 minutes of arc total.

Materials

The deviation from parallelism between two surfaces, for instance inoptical engineering,is usually measured in arcminutes or arcseconds. In addition, arcseconds are sometimes used inrocking curve(ω-scan)x ray diffractionmeasurements of high-qualityepitaxialthin films.

Manufacturing

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'soptical comparatorwill often include an option to measure in "minutes and seconds".

See also

• Gradian
• Degree (angle) § Subdivisions
• Sexagesimal § Modern usage
• Square minute
• Square second
• Steradian
• Milliradian

External links

• MOA/ milsBy Robert Simeone
• A Guide tocalculate distance using MOA Scopeby Steve Coffman