# Degree (angle)

One degree (shown in red) and
eighty nine (shown in blue)

A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of plane angle, representing 1360 of a full rotation. It is not an SI unit, as the SI unit for angles is radian, but it is mentioned in the SI brochure as an accepted unit.[1] 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.

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.[2] 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, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers. One of the earliest recorded use of 360 as the total days in a year is found in the Hebrew Bible, Genesis 8–9 as it describes the duration of the Great Flood over 12 months of 30 days each.

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.[3][4] 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 Hipparchos seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically.[5][6] Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes.[7] 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:[8]

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,[9] making it one of only 7 numbers such that no number less than twice as much has more divisors (sequence A072938 in OEIS).[10][11] Furthermore, it is divisible by every number from 1 to 10 except 7.[12] 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 latitudes and longitudes on the Earth, degree measurements may be written with decimal places: for example, 40.1875°, with the degree symbol behind the decimals.

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, and seconds instead of degrees with decimal places is sometimes referred to as DMS notation. These units, also called the arcminute and arcsecond, are respectively represented as a single and double prime: for example, 40.1875° = 40° 11′ 15″ . Sometimes single and double quotation marks are used instead: 40° 11' 15" .

If still more accuracy is required, current practice is to use decimal divisions of the second: for example, 40° 11′ 15.4″. 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[citation needed] 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. Hence the modern symbols for the minute and second of arc, and the word "second" also refer to this system.[citation needed]

## Alternative units

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° = π180.

The turn (or revolution, full circle, full rotation, cycle) is used in technology and science. 1 turn = 360°.

With the invention of the metric system, based on powers of ten, there was an attempt to define a "decimal degree" (grad or gon), so that the number of decimal degrees in a right angle would be 100 gon, and there would be 400 gon in a circle. Although this idea was abandoned already by Napoleon, some groups have continued to use it and many scientific calculators still support it. Decigrades 14000 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 16400 to 16000, each approximately equal to one milliradian. However, 16000 used by the Russian Army originated in Imperial Russia, 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 Petersberg Museum of Artillery).

### Conversion of some common angles

Units Values
Turns   0 1/24 1/12 1/10 1/8 1/6 1/5 1/4 1/3 2/5 1/2 3/4 1
Radians 0 $\tfrac{\pi}{12}$ $\tfrac{\pi}{6}$ $\tfrac{\pi}{5}$ $\tfrac{\pi}{4}$ $\tfrac{\pi}{3}$ $\tfrac{2\pi}{5}$ $\tfrac{\pi}{2}$ $\tfrac{2\pi}{3}$ $\tfrac{4\pi}{5}$ $\pi\,$ $\tfrac{3\pi}{2}$ $2\pi\,$
Degrees   15° 30° 36° 45° 60° 72° 90° 120° 144° 180° 270° 360°
Grads 0g $16{2\over 3}^g$ $33{1\over 3}^g$ 40g 50g $66{2\over 3}^g$ 80g 100g $133{1\over 3}^g$ 160g 200g 300g 400g

## Notes and references

1. ^ Bureau International des Poid et Mesures (2006). "The International System of Units (SI)" (8 ed.).
2. ^ Degree, MathWorld
3. ^ James Hopwood Jeans (1947). The Growth of Physical Science. p. 7.
4. ^ Francis Dominic Murnaghan (1946). Analytic Geometry. p. 2.
5. ^ D. Rawlins on Aristarchus
6. ^ Gerald J. Toomer. Hipparchus and Babylonian astronomy.
7. ^ 2. "Aristarchos Unbound: Ancient Vision". DIO, The International Journal of Scientific History 14: p.19. March 2008. ISSN 1041-5440. Footnote 24.
8. ^ Dirghatamas. Rigveda. pp. 1.164.48.
9. ^ 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.
10. ^ Divisibility highly composite numbers, Werner Brefeld
11. ^ Werner Brefeld (2015). (Title: not defined). Rowohlt Verlag. pp. Not yet published.
12. ^ Contrast this with the relatively unwieldy 2520, which is the least common multiple for every number from 1 to 10.