Minute of arc
Encyclopedia
A minute of arc, arcminute, or minute of angle (MOA), is a unit of angular measurement
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

equal to one sixtieth of one degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

. In turn, a second of arc or arcsecond is one sixtieth of one minute of arc.
Since one degree is defined as one three hundred and sixtieth of a rotation, 1 minute of arc is of the same. It is used in those fields which require a unit for the expression of small angles, such as astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

and marksmanship.

The number of square arcminutes in a complete sphere is

or approximately 148,510,660.498 square arcminutes.

The arcsecond is of a degree, or of a circle, or radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

.

To express even smaller angles, standard SI prefix
SI prefix
The International System of Units specifies a set of unit prefixes known as SI prefixes or metric prefixes. An SI prefix is a name that precedes a basic unit of measure to indicate a decadic multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol...

es can be employed; in particular, the milliarcsecond, abbreviated mas, is used in astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

.

## Symbols and abbreviations

The standard symbol for marking the arcminute is the prime
Prime (symbol)
The prime symbol , double prime symbol , and triple prime symbol , etc., are used to designate several different units, and for various other purposes in mathematics, the sciences and linguistics...

(′) (U+2032), though a single quote (') (U+0027) is commonly used where only ASCII
ASCII
The American Standard Code for Information Interchange is a character-encoding scheme based on the ordering of the English alphabet. ASCII codes represent text in computers, communications equipment, and other devices that use text...

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
Circumflex
The circumflex is a diacritic used in the written forms of many languages, and is also commonly used in various romanization and transcription schemes. It received its English name from Latin circumflexus —a translation of the Greek περισπωμένη...

over it ().

The standard symbol for the arcsecond is the double prime (″) (U+2033), though a double quote (") (U+0022) is commonly used where only ASCII
ASCII
The American Standard Code for Information Interchange is a character-encoding scheme based on the ordering of the English alphabet. ASCII codes represent text in computers, communications equipment, and other devices that use text...

characters are permitted. One arcsecond is thus written 1″. It is also abbreviated as arcsec or asec.
The sexagesimal system of angular measurement
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

Unit | Value | Symbol | Abbreviations | In radians (approx.)
Degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

circle °
Degree symbol
The degree symbol is a typographical symbol that is used, among other things, to represent degrees of arc or degrees of temperature...

Arcminute degree ′ (prime
Prime (symbol)
The prime symbol , double prime symbol , and triple prime symbol , etc., are used to designate several different units, and for various other purposes in mathematics, the sciences and linguistics...

)
arcmin, amin, am, , MOA 290.8882087 µrad
Arcsecond arcminute ″ (double prime) arcsec, asec, as 4.8481368 µrad
Microarcsecond 1 × 10−6 arcsecond   μas 4.8481368 prad

Celestial navigation, also known as astronavigation, is a position fixing technique that has evolved over several thousand years to help sailors cross oceans without having to rely on estimated calculations, or dead reckoning, to know their position...

, seconds of arc are rarely used in calculations, the preference usually being for degrees, minutes and decimals of a minute, written for example as 42° 25′.32 or 42° 25′.322. This notation has been carried over into marine GPS
Global Positioning System
The Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...

receivers, which normally display latitude and longitude in the latter format by default.

### Firearms

The arcminute is commonly found in the firearm
Firearm
A firearm is a weapon that launches one, or many, projectile at high velocity through confined burning of a propellant. This subsonic burning process is technically known as deflagration, as opposed to supersonic combustion known as a detonation. In older firearms, the propellant was typically...

s industry and literature, particularly concerning the accuracy of rifle
Rifle
A rifle is a firearm designed to be fired from the shoulder, with a barrel that has a helical groove or pattern of grooves cut into the barrel walls. The raised areas of the rifling are called "lands," which make contact with the projectile , imparting spin around an axis corresponding to the...

s, though the industry tends to refer to it as minute of angle. It is especially popular with shooters familiar with the Imperial measurement system because 1 MOA subtends approximately one inch
Inch
An inch is the name of a unit of length in a number of different systems, including Imperial units, and United States customary units. There are 36 inches in a yard and 12 inches in a foot...

at 100 yard
Yard
A yard is a unit of length in several different systems including English units, Imperial units and United States customary units. It is equal to 3 feet or 36 inches...

s, a traditional distance on target ranges
Shooting range
A shooting range or firing range is a specialized facility designed for firearms practice. Each facility is typically overseen by one or more supervisory personnel, called variously a range master or "RSO – Range Safety Officer" in the United States or a range conducting officer or "RCO" in the UK...

. Since most modern rifle scopes are adjustable in half , quarter , or eighth MOA increments, also known as clicks, this makes zeroing and adjustments much easier. For example, if the point of impact is 3" high and 1.5" left of the point of aim at 100 yards, 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 an MOA scale printed on them, and even figuring the right number of clicks is relatively easy on scopes that click in fractions of MOA.

One thing to be aware of is that some scopes, including some higher-end models, are calibrated such that an adjustment of 1 MOA corresponds to exactly 1 inch, rather than 1.047". 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 10 inches or more. In varmint hunting or competitive target shooting, this can easily mean the difference between a hit and a miss.

Calculating the physical equivalent group size equal to one minute of arc can be done using the equation: equivalent group size = tan × distance. In the example previously given and substituting 3,600 inches for 100 yards, 3,600 tan inches = 1.047 inches.

In metric units 1 MOA at 100 meters = 2.908 centimeters.

Sometimes, a precision firearm's accuracy will be measured in MOA. This simply means that under ideal conditions i.e. no wind, match-grade ammo, clean barrel, and 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 shooting an average 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.

Rifle manufacturers and gun magazines often refer to this capability as sub-MOA, meaning it shoots under 1 MOA. This is typically a single group of 3 to 5 shots at 100 yards, or the average of several groups. If larger samples are taken (i.e., more shots per group) then group size typically increases.

For example mathematical statistical calculation yields the following accuracy for exactly the same rifle and ammunition combination (standard deviations of every shot from center is 1 MOA):
No of shots Size (′/MOA)
2 1.77
3 2.41
5 3.07
10 3.81
20 4.45
100 5.69

### Cartography

Minutes of angle (and its subunit, seconds of angle or SOA—equal to a sixtieth of a MOA) are also used in cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

. At sea level
Sea level
Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...

, one minute of angle (around the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

or a meridian
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

) equals about 1.86 kilometres (1.2 mi)), approximately one nautical mile
Nautical mile
The nautical mile is a unit of length that is about one minute of arc of latitude along any meridian, but is approximately one minute of arc of longitude only at the equator...

(approximately, because the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

is slightly oblate); a second of angle is one sixtieth of this amount: about 30 meters or 100 feet.

Traditionally positions are given using degrees, minutes, and seconds of angles in two measurements: one for latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, the angle north or south of the equator; and one for longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

, the angle east or west of the Prime Meridian
Prime Meridian
The Prime Meridian is the meridian at which the longitude is defined to be 0°.The Prime Meridian and its opposite the 180th meridian , which the International Date Line generally follows, form a great circle that divides the Earth into the Eastern and Western Hemispheres.An international...

. Using this method, any position on or above the Earth's reference ellipsoid
Reference ellipsoid
In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body....

can be precisely given. However, because of the somewhat clumsy base-60 nature of MOA and SOA, many people now prefer to give positions using degrees only, expressed in decimal form to an equal amount of precision. Degrees, given to three decimal places ( of a degree), have about the precision as degrees-minutes-seconds ( of a degree), and so identify locations within about 120 meters or 400 feet.

Related to cartography, property boundary surveying
Surveying
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

using the metes and bounds
Metes and bounds
Metes and bounds is a system or method of describing land, real property or real estate. The system has been used in England for many centuries, and is still used there in the definition of general boundaries...

system relies on fractions of a degree to describe property lines' angles in reference to cardinal direction
Cardinal direction
The four cardinal directions or cardinal points are the directions of north, east, south, and west, commonly denoted by their initials: N, E, S, W. East and west are at right angles to north and south, with east being in the direction of rotation and west being directly opposite. Intermediate...

s. 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 feet would 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.

### Astronomy

The arcminute and arcsecond are also used in astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

. Degrees (and therefore arcminutes) are used to measure declination
Declination
In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...

, or angular distance north or south of the celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

. The arcsecond is also often used to describe parallax
Parallax
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. The term is derived from the Greek παράλλαξις , meaning "alteration"...

, due to very small parallax angles for stellar parallax
Stellar parallax
Stellar parallax is the effect of parallax on distant stars in astronomy. It is parallax on an interstellar scale, and it can be used to determine the distance of Earth to another star directly with accurate astrometry...

, and tiny angular diameters (e.g., Venus varies between 10′′ and 60′′). The parallax, proper motion
Proper motion
The proper motion of a star is its angular change in position over time as seen from the center of mass of the solar system. It is measured in seconds of arc per year, arcsec/yr, where 3600 arcseconds equal one degree. This contrasts with radial velocity, which is the time rate of change in...

and angular diameter of a star may also be written in milliarcseconds (mas), or thousandths of an arcsecond. The parsec
Parsec
The parsec is a unit of length used in astronomy. It is about 3.26 light-years, or just under 31 trillion kilometres ....

gets its name from "parallax second", for those arcseconds.

The ESA astrometric
Astrometry
Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of our Solar System and our Galaxy, the Milky...

space probe Gaia will measure star positions to 20 microarcseconds (µas). There are about 1.3 quadrillion µas in a circle. As seen from Earth, one µas is about the size of a period at the end of a sentence in the Apollo mission manuals left on the moon.

Apart from the sun, the star with the largest angular diameter
Angular diameter
The angular diameter or apparent size of an object as seen from a given position is the “visual diameter” of the object measured as an angle. In the vision sciences it is called the visual angle. The visual diameter is the diameter of the perspective projection of the object on a plane through its...

R Doradus is the name of a red giant Mira variable star in the far-southern constellation Dorado, although visually it appears more closely associated with the constellation Reticulum. Its distance from Earth is 204 ± 9 light-years...

, a red supergiant
Supergiant
Supergiants are among the most massive stars. They occupy the top region of the Hertzsprung-Russell diagram. In the Yerkes spectral classification, supergiants are class Ia or Ib . They typically have bolometric absolute magnitudes between -5 and -12...

with a diameter of 0.05 arcsecond. Because of the effects of atmospheric seeing
Astronomical seeing
Astronomical seeing refers to the blurring and twinkling of astronomical objects such as stars caused by turbulent mixing in the Earth's atmosphere varying the optical refractive index...

, ground-based telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

s will smear the image of a star to an angular diameter of about 0.5 arcsecond; in poor seeing conditions this increases to 1.5 arcseconds or even more. The dwarf planet Pluto
Pluto
Pluto, formal designation 134340 Pluto, is the second-most-massive known dwarf planet in the Solar System and the tenth-most-massive body observed directly orbiting the Sun...

has proven difficult to resolve because its angular diameter
Angular diameter
The angular diameter or apparent size of an object as seen from a given position is the “visual diameter” of the object measured as an angle. In the vision sciences it is called the visual angle. The visual diameter is the diameter of the perspective projection of the object on a plane through its...

is about 0.1 arcsecond. This is roughly equivalent to a (40 mm) ping-pong
Table tennis
Table tennis, also known as ping-pong, is a sport in which two or four players hit a lightweight, hollow ball back and forth using table tennis rackets. The game takes place on a hard table divided by a net...

ball viewed at a distance of 50 miles (80 km).

Space telescopes are not affected by the Earth's atmosphere but are diffraction limited. For example, the Hubble space telescope
Hubble Space Telescope
The Hubble Space Telescope is a space telescope that was carried into orbit by a Space Shuttle in 1990 and remains in operation. A 2.4 meter aperture telescope in low Earth orbit, Hubble's four main instruments observe in the near ultraviolet, visible, and near infrared...

can reach an angular size of stars down to about 0.1″. Techniques exist for improving seeing on the ground. Adaptive optics
Adaptive optics is a technology used to improve the performance of optical systems by reducing the effect of wavefront distortions. It is used in astronomical telescopes and laser communication systems to remove the effects of atmospheric distortion, and in retinal imaging systems to reduce the...

, for example, can produce images around 0.05 arcsecond on a 10 m class telescope.

### Human vision

In humans, 20/20 vision is the ability to resolve a spatial pattern separated by a visual angle
Visual angle
The visual angle is the angle a viewed object subtends at the eye, usually stated in degrees of arc.It also is called the object's angular size....

of one minute of arc.
A 20/20 letter subtends 5 minutes of arc total.

For raster graphics
Raster graphics
In computer graphics, a raster graphics image, or bitmap, is a data structure representing a generally rectangular grid of pixels, or points of color, viewable via a monitor, paper, or other display medium...

, Apple Inc asserts that a display of approximately 300 ppi
Pixels per inch
Pixels per inch or pixel density is a measurement of the resolution of devices in various contexts; typically computer displays, image scanners, and digital camera image sensors....

at a distance of 12 inch from one's eye, or 57 arcseconds per pixel is the maximum amount of detail that the human retina can perceive. Raymond Soneira, president of DisplayMate Technologies, however, stated that the resolution of the human retina is higher than claimed by Apple, working out to 477 ppi at 12 inch or 36 arcseconds per pixel.

### Materials

The deviation from parallelism between two surfaces, for instance in optical engineering
Optical engineering
Optical engineering is the field of study that focuses on applications of optics. Optical engineers design components of optical instruments such as lenses, microscopes, telescopes, and other equipment that utilizes the properties of light. Other devices include optical sensors and measurement...

, is usually measured in arcminutes or arcsecond.