Degree (angle)

Source: Wikipedia, the free encyclopedia.
For other units of angular measurement, see
Angular unit
.

Degree
One degree (shown in red) and eighty nine degrees (shown in blue). The lined area is a right angle.
General information
Unit systemNon-SI accepted unit
Unit ofAngle
Symbol°[1][2], deg[3]
Conversions
1°[1][2] in ...... is equal to ...
   
gons
   10/9g

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

full rotation is 360 degrees.[4]

It is not an

SI brochure as an accepted unit.[5]
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.[6]

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

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.[4]

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

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.

arc minutes.[11] Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts.[citation needed
]

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).[12] 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 (DD notation); for example, 40.1875°.

Alternatively, the traditional

arcsecond, are represented by a single prime (′) and double prime
(″) respectively. For example, 40.1875° = 40° 11′ 15″. 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. The example above would be given as 40° 11.25′ (commonly written as 11′25 or 11′.25).[13]

The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, was used by

fourth, etc.[14] Hence, the modern symbols for the minute and second of arc, and the word "second" also refer to this system.[15]

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

Alternative units

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

In most

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 (corresponding to a cycle or revolution) is used in technology and science.[citation needed
] One turn is equal to 360°.

With the invention of the

grad. Due to confusion with the existing term grad(e) in some northern European countries (meaning a standard degree, 1/360 of a turn), the new unit was called Neugrad in German (whereas the "old" degree was referred to as Altgrad), likewise nygrad in Danish, Swedish and Norwegian (also gradian), and nýgráða 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 (1⁄4,000) were used with French artillery sights in World War I.

An

St. Petersburg
Museum of Artillery.

Conversion of common angles
Turns Radians Degrees Gradians
0 turn 0 rad 0° 0g
1/72 turn π/36 rad 5° 5+5/9g
1/24 turn π/12 rad 15° 16+2/3g
1/16 turn π/8 rad 22.5° 25g
1/12 turn π/6 rad 30° 33+1/3g
1/10 turn π/5 rad 36° 40g
1/8 turn π/4 rad 45° 50g
1/2π turn 1 rad approx. 57.3° approx. 63.7g
1/6 turn π/3 rad 60° 66+2/3g
1/5 turn 2π/5 rad 72° 80g
1/4 turn π/2 rad 90° 100g
1/3 turn 2π/3 rad 120° 133+1/3g
2/5 turn 4π/5 rad 144° 160g
1/2 turn π rad 180° 200g
3/4 turn 3π/2 rad 270° 300g
1 turn 2π rad 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. 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. ^ a b Weisstein, Eric W. "Degree". mathworld.wolfram.com. Retrieved 31 August 2020.
  5. ISBN 978-92-822-2272-0
    , c. 4, pp. 145–146.
  6. ISBN 978-0-6151-7984-1. [1]
  7. Jeans, James Hopwood (1947). The Growth of Physical Science. Cambridge University Press (CUP). p. 7
    .
  8. ^ Murnaghan, Francis Dominic (1946). Analytic Geometry. p. 2.
  9. ^ Rawlins, Dennis. "On Aristarchus". DIO - the International Journal of Scientific History.
  10. Toomer, Gerald James
    . Hipparchus and Babylonian astronomy.
  11. ISSN 1041-5440. Retrieved 16 October 2015. {{cite book}}: |journal= ignored (help
    )
  12. ^ Brefeld, Werner. "Teilbarkeit hochzusammengesetzter Zahlen" [Divisibility highly composite numbers] (in German).
  13. ISBN 9781-9051-04949
    .
  14. ^ Al-Biruni (1879) [1000]. The Chronology of Ancient Nations. Translated by Sachau, C. Edward. pp. 147–149.
  15. ISBN 1-34920177-4
    .

External links

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