One degree (shown in red) and

ninety degrees (shown in blue)

*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.

## History

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.
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

chord 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

*arcminute* and

*arcsecond*, 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. These subdivisions were denoted 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.

## 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 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.

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 to , each approximately equal to one
milliradian.

## See also

## Notes

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

## External links