A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the

degree symbol The degree symbol or degree sign, , is a glyph or symbol that is used, among other things, to represent degrees of arc (e.g. in geographic coordinate systems), hours (in the medical field), degrees of temperature or alcohol proof. The symbo ...

), is a measurement of a plane

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

in which one full rotation is 360 degrees. It is not an

SI unit The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of units of measurement, system of measurement. It is the only system ...

—the SI unit of angular measure is the

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...

—but it is mentioned in the

SI brochure The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official st ...

as an accepted unit. Because a full rotation equals 2 radians, one degree is equivalent to radians.

History

Equilateral chord with length equal to radius

The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the

ecliptic The ecliptic or ecliptic plane is the orbital plane of Earth's orbit, Earth around the Sun. It was a central concept in a number of ancient sciences, providing the framework for key measurements in astronomy, astrology and calendar-making. Fr ...

path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient

calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single and specific day within such a system. A calendar is ...

s, such as the Persian calendar and the

Babylonian calendar The Babylonian calendar was a lunisolar calendar used in Mesopotamia from around the 2nd millennium BC until the Seleucid Era ( 294 BC), and it was specifically used in Babylon from the Old Babylonian Period ( 1780s BC) until the Seleucid Era. ...

, used 360 days for a year. The use of a calendar with 360 days may be related to the use of

sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...

numbers. Another theory is that the Babylonians subdivided the circle using the angle of an

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

as the basic unit, and further subdivided the latter into 60 parts following their

numeric system. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle. A chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard

divisions, was a degree.

Aristarchus of Samos Aristarchus of Samos (; , ; ) was an ancient Greek astronomer and mathematician who presented the first known heliocentric model that placed the Sun at the center of the universe, with the Earth revolving around the Sun once a year and rotati ...

and

Hipparchus Hipparchus (; , ; BC) was a Ancient Greek astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hippar ...

seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus,

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes.

Eratosthenes Eratosthenes of Cyrene (; ; – ) was an Ancient Greek polymath: a Greek mathematics, mathematician, geographer, poet, astronomer, and music theory, music theorist. He was a man of learning, becoming the chief librarian at the Library of A ...

used a simpler

system dividing a circle into 60 parts. Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

s,The divisors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. making it one of only 7 numbers such that no number less than twice as much has more divisors . Furthermore, it is divisible by every number from 1 to 10 except 7.Contrast this with the relatively unwieldy 2520, which is the

least common multiple In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ...

for every number from 1 to 10. This property has many useful applications, such as dividing the world into 24

time zone A time zone is an area which observes a uniform standard time for legal, Commerce, commercial and social purposes. Time zones tend to follow the boundaries between Country, countries and their Administrative division, subdivisions instead of ...

s, each of which is nominally 15° of

longitude Longitude (, ) is a geographic coordinate that specifies the east- west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lett ...

, to correlate with the established 24-hour

day A day is the time rotation period, period of a full Earth's rotation, rotation of the Earth with respect to the Sun. On average, this is 24 hours (86,400 seconds). As a day passes at a given location it experiences morning, afternoon, evening, ...

convention. Finally, it may be the case that more than one of these factors has come into play. According to that theory, the number is approximately 365 because of the apparent movement of the sun against the celestial sphere, and that it was rounded to 360 for some of the mathematical reasons cited above.

Subdivisions

For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in

astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

or for geographic coordinates (

latitude In geography, latitude is a geographic coordinate system, geographic coordinate that specifies the north-south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at t ...

and

), degree measurements may be written using decimal degrees (''DD notation''); for example, 40.1875°. Alternatively, the traditional

unit subdivisions can be used: one degree is divided into 60 ''minutes (of arc)'', and one minute into 60 ''seconds (of arc)''. Use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the ''

arcminute A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...

'' and ''

arcsecond A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a tu ...

'', are represented by a single prime (′) and

double prime The prime symbol , double prime symbol , triple prime symbol , and quadruple prime symbol are used to designate units and for other purposes in mathematics, science, linguistics and music. Although the characters differ little in appearance fr ...

(″) respectively. For example, . Additional precision can be provided using decimal fractions of an arcsecond. Maritime charts are marked in degrees and decimal minutes to facilitate measurement; 1 minute of latitude is 1

nautical mile A nautical mile is a unit of length used in air, marine, and space navigation, and for the definition of territorial waters. Historically, it was defined as the meridian arc length corresponding to one minute ( of a degree) of latitude at t ...

. The example above would be given as 40° 11.25′ (commonly written as 11′25 or 11′.25). The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today. These subdivisions were denoted by writing the

Roman numeral Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ea ...

for the number of sixtieths in superscript: 1^I for a "

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

" (minute of arc), 1^II for a

second The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of U ...

, 1^III for a third, 1^IV for a fourth, etc. Hence, the modern symbols for the minute and second of arc, and the word "second" also refer to this system.

SI prefixes A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pre ...

can also be applied as in, e.g., millidegree, microdegree, etc.

Alternative units

In most

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

work beyond practical geometry, angles are typically measured in

s rather than degrees. This is for a variety of reasons; for example, the

trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

s have simpler and more "natural" properties when their arguments are expressed in radians. These considerations outweigh the convenient divisibility of the number 360. One complete turn (360°) is equal to 2'' '' radians, so 180° is equal to radians, or equivalently, the degree is a

mathematical constant A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathem ...

: 1° = . One turn (corresponding to a cycle or revolution) is equal to 360°. With the invention of the

metric system The metric system is a system of measurement that standardization, standardizes a set of base units and a nomenclature for describing relatively large and small quantities via decimal-based multiplicative unit prefixes. Though the rules gover ...

, based on powers of ten, there was an attempt to replace degrees by decimal "degrees" in France and nearby countries,These new and decimal "degrees" must not be confused with decimal degrees. where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° = gon). This was called or '' grad''. Due to confusion with the existing term ''grad(e)'' in some northern European countries (meaning a standard degree, of a turn), the new unit was called in German (whereas the "old" degree was referred to as ), likewise in Danish, Swedish and Norwegian (also ''gradian''), and in Icelandic. To end the confusion, the name ''gon'' was later adopted for the new unit. Although this idea of metrification was abandoned by Napoleon, grades continued to be used in several fields and many scientific calculators support them. Decigrades () were used with French artillery sights in World War I. An angular mil, which is most used in military applications, has at least three specific variants, ranging from to . It is approximately equal to one

milliradian A milliradian (International System of Units, SI-symbol mrad, sometimes also abbreviated mil) is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Milliradians are used in adjustment of ...

( ). A mil measuring of a revolution originated in the

imperial Russian army The Imperial Russian Army () was the army of the Russian Empire, active from 1721 until the Russian Revolution of 1917. It was organized into a standing army and a state militia. The standing army consisted of Regular army, regular troops and ...

, where an equilateral chord was divided into tenths to give a circle of 600 units. This may be seen on a lining plane (an early device for aiming

indirect fire Indirect fire is aiming and firing a projectile without relying on a direct line of sight between the gun and its target, as in the case of direct fire. Aiming is performed by calculating azimuth and inclination, and may include correcting ...

artillery) dating from about 1900 in the St. Petersburg Museum of Artillery.

Notes

References

External links