Degree (angle)

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.[1]

Contents

History

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

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. "In addition to being divisible by 24, 360 is divisible by 15 and hence corresponds with 24 time zones. Each time zone has an approximate value of 15 degrees of longitude. 360 degrees has been part of mans' primary circular navigational system due to time relativity / compatibility -- 24 hours in 1 earth rotation." ^3

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.

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: 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.

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. 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.

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

Conversion of some common angles

Units Values
Turns   0 1/12 1/8 1/6 1/4 1/2 3/4 1
Degrees   0° 30° 45° 60° 90° 180° 270° 360°
Radians 0 \frac{\pi}{6} \frac{\pi}{4} \frac{\pi}{3} \frac{\pi}{2} \pi\, \frac{3\pi}{2} 2\pi\,
Grads 0g 33⅓g 50g 66⅔g 100g 200g 300g 400g

See also

Notes

  1. ↑ Beckmann P. (1976) A History of Pi, St. Martin's Griffin. ISBN 0-312-38185-9
  2. ↑ Degree, MathWorld
 3.^ SSG. Bernard Klein 82C US ARMY (Ret.) Carthage College, Kenosha, WI.

External links