# Degree (angle)

*This article describes the unit of angle. For other meanings, see degree.*

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 ^{1}/_{360} of a full rotation; one degree is equivalent to π/180 radians. When that angle is with respect to a reference meridian, it indicates a location along a great circle of a sphere, such as Earth (see Geographic coordinate system), Mars, or the celestial sphere.^{[1]}

## History

File:Equilateral chord.svg
The selection of 360 as the number of degrees (*i.e.,* smallest practical sub-arcs) in a circle was probably based on the fact that 360 is approximately the number of days in a year. Its use is often said to originate from the methods of the ancient Babylonians.^{[2]} Ancient astronomers noticed that the stars in the sky, which circle the celestial pole every day, seem to advance in that circle by approximately one-360th of a circle, *i.e.,* one degree, each day. (Ancient calendars, such as the Persian Calendar, used 360 days for a year.) Its application to measuring angles in geometry can possibly be traced to Thales who popularized geometry among the Greeks and lived in Anatolia (modern western Turkey) among people who had dealings with Egypt and Babylon.

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; while six such chords completed the full circle.

Another motivation for choosing the number 360 is that it is readily divisible: 360 has 24 divisors (including 1 and 360), including every number from 1 to 10 except 7. For the number of degrees in a circle to be divisible by every number from 1 to 10, there would need to be 2520 degrees in a circle, which is a much less convenient number.

- 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.

### India

The division of the circle into 360 parts also occurred in ancient India, as evidenced in the Rig Veda:

- 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.
- (Dirghatama, Rig Veda 1.164.48)

## 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, but the traditional sexagesimal unit subdivision is commonly seen. One degree is divided into 60 *minutes (of arc)*, and one minute into 60 *seconds (of arc)*. These units, also called the *arc minute* and *arc second*, are respectively represented as a single and double prime, or if necessary by a single and double quotation mark: for example, 40.1875° = 40° 11′ 15″ (or 40° 11' 15").

If still more accuracy is required, current practice is to use decimal divisions of the second. The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, is rarely used today.^{[citation needed]} These subdivisions were denoted^{[citation needed]} by writing the Roman numeral for the number of sixtieths in superscript: 1^{I} for a "prime" (minute of arc), 1^{II} for a second, 1^{III} for a third, 1^{IV} for a fourth, etc. Hence the modern symbols for the minute and second of arc.^{[citation needed]}

## Alternative units

*See also: Measuring angles.*

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 2*π* radians, so 180° is equal to π radians, or equivalently, the degree is a mathematical constant 1° = ^{π}⁄_{180}.

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 did not gain much momentum, most scientific calculators used to support it.^{[citation needed]}

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

An angular mil, which is most used in military applications, has at least three specific variants, ranging from ^{1}⁄_{6000} to ^{1}⁄_{6400}, each approximately equal to one milliradian.

## See also

- Gradian
- Square degree
- Steradian
- Compass
- Geographic coordinate system

## Notes

- ↑ Beckmann P. (1976)
*A History of Pi*, St. Martin's Griffin. ISBN 0-312-38185-9 - ↑ Degree, MathWorld

## External links

- Degrees as an angle measure, with interactive animation
- Degree at MathWorld