Degree (angle)
From Exampleproblems
- 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).
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History
The number 360 was probably adopted because of the number of days in a year. Primitive calendars, such as the Persian Calendar used 360 days for a year. This was most likely due to watching stars revolve around the North Star forming a circle one degree per day. 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 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 ').
Further justification
360 is also 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: 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.
Radians
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 (symbol rad, an angle corresponding to an arc of a circle whose length equals the circle's radius) as opposed to its radius of curvature, or arcradius. Thus 180° = π rad, 1° ≈ 0.0174533 rad, and 1 rad ≈ 57.29578°. The radian is also the SI unit of angle.
Metric "decimal" degree
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, and there would be 400 degrees in a circle. Although this idea did not gain much momentum, most scientific calculators still support it.
