Misplaced Pages

This is an old revision of this page, as edited by Crum375 (talk | contribs) at 22:38, 22 September 2006 (→History: removed PI derivation as I don't see how this explains the historic derivation of the 'degree', 1/360 of the circle, which is this article). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

This article describes "degree" as a unit of angle. For alternative meanings, see degree.

A degree (in full, a degree of arc, arc degree, or arcdegree), usually symbolized °, is a measurement of plane angle, representing 1／360 of a full rotation. When that angle is with respect to a reference meridian, it indicates a location along a great circle of a sphere (such as Earth, Mars, or the celestial sphere).

The degree and its subdivisions are the only units in use which are written without a separating space between the number and unit symbol (e.g. 15° 30', not 15 ° 30 ').

History

The number 360 as the number of 'degrees' (or minimal/practical sub-arcs) in a circle, and hence the unit of a degree as a sub-arc of 1/360 of the circle, was probably adopted because it approximates the number of days in a year. 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. Primitive 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.

Further justification

The number 360 is useful since 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.

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 closing quotation mark: for example, 40.1875° = 40° 11' 15". If still more accuracy is required, decimal divisions of the second are normally used, rather than thirds of 1／60 second, fourths of 1／60 of a third, and so on. These (rarely used) subdivisions were noted by writing the Roman numeral for the number of sixtieths in superscript: 1 for a "prime" (minute of arc), 1 for a second, 1 for a third, 1 for a fourth, etc. Hence the modern symbols for the minute and second of arc.

Alternative units

In mathematics, angles in degrees are rarely used, as the convenient divisibility of the number 360 is not so important. For various reasons, mathematicians typically prefer to use the radian. In this system the angles 180° and π radians are equal, or equivalently, the degree is a mathematical constant ° = π/180. This means, that in a complete circle (360°) there are 2π radians. The circumference of a circle is 2πr, where r is the radius.

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 still support it.

An angular mil is a 1/1000 of a radian, which is convenient for survey and distance estimations due to simple trigonometry.

See also

• Radian
• Gradian
• Square degree
• Steradian
• Compass

References

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

External links

• Degrees as an angle measure, with interactive animation

• Units of angle
• Imperial units
• Mathematical constants
• Customary units in the United States