Degree (angle)

A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol),[4] is a measurement of a plane angle in which one full rotation is 360 degrees.[5]

For other units of angular measurement, see Angular unit.
Degree
Unit systemNon-SI accepted unit
Unit ofAngle
Symbol°[1][2] or deg[3]
Conversions
1 °[1][2] in ...... is equal to ...
   turns   1/360 turn
   radians   π/180 rad ≈ 0.01745.. rad
   milliradians   50·π/9 mrad ≈ 17.45.. mrad
   gons   10/9g
One degree (shown in red) and
eighty nine degrees (shown in blue)

It is not an SI unit—the SI unit of angular measure is the radian—but it is mentioned in the SI brochure as an accepted unit.[6] Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians.

History
A circle with an equilateral chord (red). One sixtieth of this arc is a degree. Six such chords complete the circle.[7]

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.[5] Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Persian calendar and the Babylonian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers.

Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit, and further subdivided the latter into 60 parts following their sexagesimal numeric system.[8][9] 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 sexagesimal divisions, was a degree.

Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically.[10][11] Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes.[12] Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts.

The division of the circle into 360 parts also occurred in ancient India, as evidenced in the Rigveda:[13]

Twelve spokes, one wheel, navels three.

Who can comprehend this?
On it are placed together
three hundred and sixty like pegs.
They shake not in the least.

— Dirghatamas, Rigveda 1.164.48

Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24 divisors,[note 1] making it one of only 7 numbers such that no number less than twice as much has more divisors (sequence A072938 in the OEIS).[14][15] Furthermore, it is divisible by every number from 1 to 10 except 7.[note 2] This property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day 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 or for geographic coordinates (latitude and longitude), degree measurements may be written using decimal degrees, with the degree symbol behind the decimals; for example, 40.1875°.

Alternatively, the traditional sexagesimal 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 and arcsecond, are respectively represented by a single prime (′) and a double prime (″).[4] For example, 40.1875° = 40° 11′ 15″, or, using quotation mark characters, 40° 11' 15". Additional precision can be provided using decimals for the arcseconds component.

Maritime charts are marked in degrees and decimal minutes to facilitate measurement; 1 minute of latitude is 1 nautical mile. The example above would be given as 40° 11.25′ (commonly written as 11′25 or 11′.25).[16]

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 for the number of sixtieths in superscript: 1I for a "prime" (minute of arc), 1II for a second, 1III for a third, 1IV for a fourth, etc.[17] Hence, the modern symbols for the minute and second of arc, and the word "second" also refer to this system.[18]

Alternative units
See also: Measuring angles
A chart to convert between degrees and radians

In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions 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 2π radians, so 180° is equal to π radians, or equivalently, the degree is a mathematical constant: 1° = ​π⁄180.

The turn (or revolution, full circle, full rotation, cycle) is used in technology and science. One turn is equal to 360°.

With the invention of the metric system, based on powers of ten, there was an attempt to replace degrees by decimal "degrees"[note 3] called grad or gon, where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° = ​10⁄9 gon). Although that idea was abandoned by Napoleon, grades continued to be used in several fields and many scientific calculators support them. Decigrades (​1⁄4,000) 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 ​1⁄6,400 to ​1⁄6,000. It is approximately equal to one milliradian (c. ​1⁄6,283). A mil measuring ​1⁄6,000 of a revolution originated in the imperial Russian army, 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 artillery) dating from about 1900 in the St. Petersburg Museum of Artillery.

Conversion of common angles
Turns Radians Degrees Gradians, or gons
0 0 0° 0g
1/24 π/12 15° 16+2/3g
1/12 π/6 30° 33+1/3g
1/10 π/5 36° 40g
1/8 π/4 45° 50g
1/2π 1 c. 57.3° c. 63.7g
1/6 π/3 60° 66+2/3g
1/5 2π/5 72° 80g
1/4 π/2 90° 100g
1/3 2π/3 120° 133+1/3g
2/5 4π/5 144° 160g
1/2 π 180° 200g
3/4 3π/2 270° 300g
1 2π 360° 400g

See also

Notes
  1. 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.
  2. Contrast this with the relatively unwieldy 2520, which is the least common multiple for every number from 1 to 10.
  3. These new and decimal "degrees" must not be confused with decimal degrees.

References
  1. HP 48G Series – User's Guide (UG) (8 ed.). Hewlett-Packard. December 1994 [1993]. HP 00048-90126, (00048-90104). Retrieved 6 September 2015.
  2. HP 50g graphing calculator user's guide (UG) (1 ed.). Hewlett-Packard. 1 April 2006. HP F2229AA-90006. Retrieved 10 October 2015.
  3. HP Prime Graphing Calculator User Guide (UG) (PDF) (1 ed.). Hewlett-Packard Development Company, L.P. October 2014. HP 788996-001. Archived from the original (PDF) on 3 September 2014. Retrieved 13 October 2015.
  4. "Compendium of Mathematical Symbols". Math Vault. 1 March 2020. Retrieved 31 August 2020.
  5. Weisstein, Eric W. "Degree". mathworld.wolfram.com. Retrieved 31 August 2020.
  6. Bureau international des poids et mesures, Le Système international d’unités (SI) / The International System of Units (SI), 9th ed. (Sèvres: 2019), ISBN 978‑92‑822‑2272‑0 Parameter error in {{ISBN}}: Invalid ISBN., c. 4, pp. 145–146.
  7. Euclid (2008). "Book 4". Euclid's Elements of Geometry [Euclidis Elementa, editit et Latine interpretatus est I. L. Heiberg, in aedibus B. G. Teubneri 1883–1885]. Translated by Heiberg, Johan Ludvig; Fitzpatrick, Richard (2 ed.). Princeton University Press. ISBN 978-0-6151-7984-1.
  8. Jeans, James Hopwood (1947). The Growth of Physical Science. Cambridge University Press (CUP). p. 7.
  9. Murnaghan, Francis Dominic (1946). Analytic Geometry. p. 2.
  10. Rawlins, Dennis. "On Aristarchus". DIO - the International Journal of Scientific History.
  11. Toomer, Gerald James. Hipparchus and Babylonian astronomy.
  12. "2 (Footnote 24)" (PDF). Aristarchos Unbound: Ancient Vision / The Hellenistic Heliocentrists' Colossal Universe-Scale / Historians' Colossal Inversion of Great & Phony Ancients / History-of-Astronomy and the Moon in Retrograde!. DIO - the International Journal of Scientific History. 14. March 2008. p. 19. ISSN 1041-5440. Retrieved 16 October 2015.
  13. Dirghatamas. Rigveda. p. 1.164.48.
  14. Brefeld, Werner. "Teilbarkeit hochzusammengesetzter Zahlen" [Divisibility highly composite numbers] (in German).
  15. Brefeld, Werner (2015). (unknown). Rowohlt Verlag. Cite uses generic title (help)
  16. Hopkinson, Sara (2012). RYA day skipper handbook - sail. Hamble: The Royal Yachting Association. p. 76. ISBN 9781-9051-04949.
  17. Al-Biruni (1879) [1000]. The Chronology of Ancient Nations. Translated by Sachau, C. Edward. pp. 147–149.
  18. Flegg, Graham H. (1989). Numbers Through the Ages. Macmillan International Higher Education. pp. 156–157. ISBN 1-34920177-4.
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