Definitions

# Minute of arc

A minute of arc, arcminute, or MOA is a unit of angular measurement, equal to one sixtieth (1/60) of one degree. Since one degree is defined as one three hundred sixtieth (1/360) of a circle, 1 MOA is 1/21600 of the amount of arc in a closed circle. It is used in those fields which require a unit for the expression of small angles, such as astronomy or marksmanship.

## Symbols, abbreviations and subdivisions

The standard 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 written 1′ (or 1'). It is also abbreviated as arcmin or amin or, less commonly, the prime with a circumflex over it ($hat\left\{\text{'}\right\}$).

The subdivision of the minute of arc is the second of arc, or arcsecond. There are 60 arcseconds in an arcminute. Therefore, the arcsecond is 1/1296000 of a circle, or (π/648000) radians, which is approximately 1/206265 radian. The symbol for the arcsecond is the double prime (″) (U+2033). To express even smaller angles, standard SI prefixes can be employed; in particular, the milliarcsecond, abbreviated mas, is sometimes used in astronomy.

The sexagesimal system of angular measurement
unit value symbol abbreviations conversion
degree (angle) 1/360 circle ° deg 17.4532925 mrad
arcminute 1/60 degree ′ (prime) arcmin, amin, $hat\left\{\text{'}\right\}$, MOA 290.8882087 µrad
arcsecond 1/60 arcminute ″ (double prime) arcsec 4.8481368 µrad
milliarcsecond 1/1000 arcsecond mas 4.8481368 nrad

## Uses

### Firearms

This unit is commonly found in the firearms industry and literature, particularly that concerning the accuracy of rifles. The industry tends to refer to it as minute of angle rather than minute of arc. It is popular because 1 MOA subtends approximately one inch at 100 yards, a traditional distance on target ranges. A shooter can easily readjust his rifle scope by measuring the distance in inches the bullet hole is from the desired impact point, and adjusting the scope that many MOA in the same direction. Most target scopes designed for long distances are adjustable in quarter (¼) or eighth (⅛) MOA "clicks". One eighth MOA is equal to approximately an eighth of an inch at 100 yards or one inch at 800 yards.

Calculating the physical equivalent group size equal to one minute of arc can be done using the equation: equivalent group size = tan(MOA ∕ 60)*distance. In the example previously given and substituting 3600 inches for 100 yards, tan(1 MOA ∕ 60)∙ 3600 inches = 1.0471975511966 inches.

In metric units 1 MOA at 100 meters = 2.90888208665722 centimeters.

Sometimes, a firearm's accuracy will be measured in MOA. This simply means that under ideal conditions, the gun is capable of repeatedly producing a group of shots whose center points (center-to-center) fit into a circle, the diameter of which can be subtended by that amount of arc. (E.g.: a "1 MOA rifle" should be capable, under ideal conditions, of shooting a 1-inch group at 100 yards, a "2 MOA rifle" a 2-inch group at 100 yards, etc.) Some manufacturers such as Weatherby and Cooper offer actual guarantees of real-world MOA performance.

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.

### Cartography

Minutes of angle (and its subunit, seconds of angle or SOA—equal to a sixtieth of a MOA) are also used in cartography and navigation. At sea level, one minute of angle (around the equator or a meridian) equals about 1.86 km or 1.15 miles, approximately one nautical mile (approximately, because the Earth 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, the angle north or south of the equator; and one for longitude, the angle east or west of the Prime Meridian. Using this method, any position on or above the Earth's reference ellipsoid 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 (1/1000 of a degree), have about 1/4 the precision as degrees-minutes-seconds (1/3600 of a degree), and so identify locations within about 120 meters or 400 feet.

### Property surveying

Related to cartography, property boundary surveying using the metes and bounds system relies on fractions of a degree to describe property lines' angles in reference to cardinal 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 feet would describe a property 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. Degrees (and therefore arcminutes) are used to measure declination, or angular distance north or south of the celestial equator. The arcsecond is also often used to describe parallax, due to very small parallax angles, and tiny angular diameters (e.g. Venus varies between 10″ and 60″). The parallax, proper motion and angular diameter of a star may also be written in milliarcseconds (mas), or thousandths of an arcsecond. The parsec gets its name from "parallax second", for those arcseconds.

Apart from the sun, the star with the largest angular diameter from Earth is R Doradus, a red supergiant with a diameter of 0.05 arcseconds. Due to the effects of atmospheric seeing, ground-based telescopes 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.

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, for example adaptive optics, which can give 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 of one minute of arc.

## References

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