Difference between revisions of "Degree (angle)"
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[[Image:Degree diagram.svg|thumb|One degree (shown in red) and<br/>eighty nine (shown in blue)]] | [[Image:Degree diagram.svg|thumb|One degree (shown in red) and<br />eighty nine (shown in blue)]] | ||
A '''degree''' (in full, a '''degree of arc''', '''arc degree''', or '''arcdegree'''), usually denoted by '''°''' (the [[degree symbol]]), is a measurement of [[plane (mathematics)|plane]] [[angle]], representing <sup>1</sup>⁄<sub>360</sub> of a [[Turn (geometry)|full rotation]] | A '''degree''' (in full, a '''degree of arc''', '''arc degree''', or '''arcdegree'''), usually denoted by '''°''' (the [[degree symbol]]), is a measurement of [[plane (mathematics)|plane]] [[angle]], representing <sup>1</sup>⁄<sub>360</sub> of a [[Turn (geometry)|full rotation]]. It is not an [[SI unit]], as the SI unit for angles is [[radian]], but it is mentioned in the SI brochure as an [[Non-SI units mentioned in the SI|accepted unit]].<ref>http://www.bipm.org/en/si/si_brochure/chapter4/table6.html</ref> Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians. | ||
==History== | ==History== | ||
[[Image:Equilateral chord.svg|thumb||right|A circle with an equilateral [[Chord (geometry)|chord]] (red). One sixtieth of this arc is a degree. Six such chords complete the circle.]] | [[Image:Equilateral chord.svg|thumb||right|A circle with an equilateral [[Chord (geometry)|chord]] (red). One sixtieth of this arc is a degree. Six such chords complete the circle.]] | ||
The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year.<ref>[http://mathworld.wolfram.com/Degree.html Degree], MathWorld</ref> Ancient [[astronomers]] noticed that the sun, | The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year.<ref>[http://mathworld.wolfram.com/Degree.html Degree], MathWorld</ref> Ancient [[astronomers]] noticed that the sun, which follows through the [[ecliptic]] path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient [[calendar]]s, such as the [[Iranian calendar|Persian calendar]], used 360 days for a year. The use of a calendar with 360 days may be related to the use of [[sexagesimal]] numbers. One of the earliest recorded use of 360 as the total days in a year is found in the Hebrew Bible, Genesis 8-9 as it describes the duration of the Great Flood over 12 months of 30 days each. | ||
Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit and further subdivided the latter into 60 parts following their [[sexagesimal]] numeric system.<ref>[[J.H. Jeans]] (1947), ''The Growth of Physical Science'', [http://books.google.co.uk/books?hl=en&lr=&id=JX49AAAAIAAJ&oi=fnd&pg=PA7 p.7]; [[Francis Dominic Murnaghan (mathematician)|Francis Dominic Murnaghan]] (1946), ''Analytic Geometry'', p.2</ref> The [[history of trigonometry|earliest trigonometry]], used by the [[Babylonian astronomy|Babylonian astronomers]] and their [[Greek astronomy|Greek]] successors, was based on [[chord (geometry)| | Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit and further subdivided the latter into 60 parts following their [[sexagesimal]] numeric system.<ref>[[J.H. Jeans]] (1947), ''The Growth of Physical Science'', [http://books.google.co.uk/books?hl=en&lr=&id=JX49AAAAIAAJ&oi=fnd&pg=PA7 p.7]; [[Francis Dominic Murnaghan (mathematician)|Francis Dominic Murnaghan]] (1946), ''Analytic Geometry'', p.2</ref> The [[history of trigonometry|earliest trigonometry]], used by the [[Babylonian astronomy|Babylonian astronomers]] and their [[Greek astronomy|Greek]] successors, was based on [[chord (geometry)|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. | ||
[[Aristarchus of Samos]] and [[Hipparchus|Hipparchos]] seem to have been among the first [[Greek astronomy|Greek scientists]] to exploit Babylonian astronomical knowledge and techniques systematically.<ref>For more information see [http://www.dioi.org/cot.htm#dqsr D.Rawlins] on Aristarchus; and [[G. J. Toomer]], "Hipparchus and Babylonian astronomy."</ref> [[Timocharis]], Aristarchus, [[Aristillus]], [[Archimedes]], and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 [[arc minute]]s ([http://www.dioi.org/vols/we0.pdf DIO 14] ‡2 p. 19 n.24). [[Eratosthenes]] used a simpler [[sexagesimal]] system dividing a circle into 60 parts. | [[Aristarchus of Samos]] and [[Hipparchus|Hipparchos]] seem to have been among the first [[Greek astronomy|Greek scientists]] to exploit Babylonian astronomical knowledge and techniques systematically.<ref>For more information see [http://www.dioi.org/cot.htm#dqsr D.Rawlins] on Aristarchus; and [[G. J. Toomer]], "Hipparchus and Babylonian astronomy."</ref> [[Timocharis]], Aristarchus, [[Aristillus]], [[Archimedes]], and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 [[arc minute]]s ([http://www.dioi.org/vols/we0.pdf DIO 14] ‡2 p. 19 n.24). [[Eratosthenes]] used a simpler [[sexagesimal]] system dividing a circle into 60 parts. | ||
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{{quotation | {{quotation | ||
|Twelve spokes, one wheel, navels three.<br/> | |Twelve spokes, one wheel, navels three.<br /> | ||
Who can comprehend this? <br/> | Who can comprehend this? <br /> | ||
On it are placed together <br/> | On it are placed together <br /> | ||
three hundred and sixty like pegs. <br/> | three hundred and sixty like pegs. <br /> | ||
They shake not in the least. | They shake not in the least. | ||
|[[Dirghatamas]] | |[[Dirghatamas]] | ||
|Rigveda 1.164.48}} | |Rigveda 1.164.48}} | ||
Another motivation for choosing the number 360 may have been that it is [[highly composite number|readily divisible]]: 360 has 24 [[divisor]]s, making it one of only 7 numbers that | Another motivation for choosing the number 360 may have been that it is [[highly composite number|readily divisible]]: 360 has 24 [[divisor]]s,<ref>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.</ref> making it one of only 7 numbers such that no number less than twice as much has more divisors {{OEIS|id=A072938}}.<ref>[http://translate.google.com/translate?sl=auto&tl=en&js=n&prev=_t&hl=en&ie=UTF-8&layout=2&eotf=1&u=http%3A%2F%2Fwww.brefeld.homepage.t-online.de%2Fteilbarkeit.html&act=url Divisibility highly composite numbers], Werner Brefeld</ref> Furthermore, it is divisible by every number from 1 to 10 except 7.<ref>Contrast this with the relatively unwieldy 2520, which is the [[least common multiple]] for every number from 1 to 10.</ref> This property has many useful applications, such as dividing the world into 24 [[time zone]]s, each of which is nominally 15° of [[longitude]], to correlate with the established [[hour|24-hour]] [[day]] convention. | ||
Finally, it may be the case that more than one of these factors has come into play. According to that theory, the number is approximately 365 because of the apparent movement of the sun against the celestial sphere and that it was rounded to 360 for some of the mathematical reasons cited above. | Finally, it may be the case that more than one of these factors has come into play. According to that theory, the number is approximately 365 because of the apparent movement of the sun against the celestial sphere and that it was rounded to 360 for some of the mathematical reasons cited above. | ||
==Subdivisions== | ==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 [[latitude]]s and [[longitude]]s on the Earth, degree measurements may be written with [[decimal]] places | 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 [[latitude]]s and [[longitude]]s on the Earth, degree measurements may be written with [[decimal]] places: for example, 40.1875°, with the degree symbol behind the decimals. | ||
Alternatively, the traditional [[sexagesimal]] [[Units of measurement|unit]] | Alternatively, the traditional [[sexagesimal]] [[Units of measurement|unit]] subdivisions can be used. 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 (symbol)|prime]]: for example, 40.1875° = 40° 11′ 15″ . Sometimes single and double quotation marks are used instead: 40° 11' 15" . | ||
If still more accuracy is required, current practice is to use decimal divisions of the second | If still more accuracy is required, current practice is to use decimal divisions of the second: for example, 40° 11′ 15.4″ . The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, was used by [[al-Kashi]] and other ancient astronomers, but is rarely used today. These subdivisions were denoted{{Citation needed|date=January 2008}} by writing the [[Roman numeral]] for the number of sixtieths in superscript: 1<sup>I</sup> for a "prime" (minute of arc), 1<sup>II</sup> for a second, 1<sup>III</sup> for a third, 1<sup>IV</sup> for a fourth, etc. Hence the modern symbols for the minute and second of arc, and the word "second" also refer to this system. {{Citation needed|date=April 2009}} | ||
==Alternative units== | ==Alternative units== | ||
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The [[Turn (geometry)|turn]] (or revolution, full circle, full rotation, cycle) is used in technology and science. 1 turn = 360°. | The [[Turn (geometry)|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 (angle)|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 [[calculator]]s still support it. | With the invention of the [[metric system]], based on powers of ten, there was an attempt to define a "decimal degree" ('''[[grad (angle)|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 [[calculator]]s still support it. Decigrades {{frac|4000}} were used with French artillery sights in World War I. | ||
An [[angular mil]], which is most used in military applications, has at least three specific variants, ranging from {{frac|6400}} to {{frac|6000}}, each approximately equal to one milliradian. However, {{frac|6000}} used by the Russian Army originated in Imperial Russia, where an equilateral chord was divided into tenths to give a circle of 600 units (this may be seen on a | An [[angular mil]], which is most used in military applications, has at least three specific variants, ranging from {{frac|6400}} to {{frac|6000}}, each approximately equal to one milliradian. However, {{frac|6000}} used by the Russian Army originated in Imperial Russia, where an equilateral chord was divided into tenths to give a circle of 600 units (this may be seen on a dial sight, a device for accurate aiming of artillery, also known as an aiming circle, dating from about 1900, in the St Petersberg Museum of Artillery). | ||
===Conversion of some common angles=== | ===Conversion of some common angles=== | ||
{|class = wikitable | {|class = wikitable style="text-align:center;" | ||
! Units !! colspan= | ! Units !! colspan=13 | Values | ||
|- | |- | ||
|style = "background:#f2f2f2" | '''[[Turn (geometry)|Turns]]''' | |style = "background:#f2f2f2; text-align:left;" | '''[[Turn (geometry)|Turns]]''' | ||
|style = "width:3em; | |style = "width:3em;" | 0 | ||
|style = "width:3em; | |style = "width:3em;" | 1/24 | ||
|style = "width:3em; | |style = "width:3em;" | 1/12 | ||
|style = "width:3em; | |style = "width:3em;" | 1/10 | ||
|style = "width:3em; | |style = "width:3em;" | 1/8 | ||
|style = "width:3em; | |style = "width:3em;" | 1/6 | ||
|style = "width:3em; | |style = "width:3em;" | 1/5 | ||
|style = "width:3em; | |style = "width:3em;" | 1/4 | ||
|style = "width:3em;" | 1/3 | |||
|style = "width:3em;" | 2/5 | |||
|style = "width:3em; | |style = "width:3em;" | 1/2 | ||
|style = "width:3em; | |style = "width:3em;" | 3/4 | ||
|style = "width:3em; | |style = "width:3em;" | 1 | ||
|style = "width:3em; | |- | ||
|style = "width:3em; | |style = "background:#f2f2f2; text-align:left;" | '''[[Radian]]s''' | ||
| | | 0 | ||
|style = " | | <math>\tfrac{\pi}{12}</math> | ||
| <math>\tfrac{\pi}{6}</math> | |||
| <math>\tfrac{\pi}{5}</math> | |||
| <math>\tfrac{\pi}{4}</math> | |||
| <math>\tfrac{\pi}{3}</math> | |||
| | | <math>\tfrac{2\pi}{5}</math> | ||
| | | <math>\tfrac{\pi}{2}</math> | ||
| <math>\tfrac{2\pi}{3}</math> | |||
| | | <math>\tfrac{4\pi}{5}</math> | ||
| | | <math>\pi\,</math> | ||
| | | <math>\tfrac{3\pi}{2}</math> | ||
| <math>2\pi\,</math> | |||
|- | |- | ||
|style = "background:#f2f2f2" | '''[[Grad (angle)|Grads]]''' | |style = "background:#f2f2f2; text-align:left;" | '''Degrees''' | ||
|style = "width:3em;" | 0° | |||
| | |style = "width:3em;" | 15° | ||
|style = "width:3em;" | 30° | |||
| | |style = "width:3em;" | 36° | ||
|style = "width:3em;" | 45° | |||
| | |style = "width:3em;" | 60° | ||
|style = "width:3em;" | 72° | |||
|style = "width:3em;" | 90° | |||
|style = "width:3em;" | 120° | |||
|style = "width:3em;" | 144° | |||
|style = "width:3em;" | 180° | |||
|style = "width:3em;" | 270° | |||
|style = "width:3em;" | 360° | |||
|- | |||
|style = "background:#f2f2f2; text-align:left;" | '''[[Grad (angle)|Grads]]''' | |||
| 0<sup>g</sup> | |||
| <math>16{2\over 3}^g</math> | |||
| <math>33{1\over 3}^g</math> | |||
| 40<sup>g</sup> | |||
| 50<sup>g</sup> | |||
| <math>66{2\over 3}^g</math> | |||
| 80<sup>g</sup> | |||
| 100<sup>g</sup> | |||
| <math>133{1\over 3}^g</math> | |||
| 160<sup>g</sup> | |||
| 200<sup>g</sup> | |||
| 300<sup>g</sup> | |||
| 400<sup>g</sup> | |||
|} | |} | ||
==See also== | ==See also== | ||
*[[ | * [[Compass]] | ||
*[[ | * [[Geographic coordinate system]] | ||
*[[ | * [[Gradian]] | ||
*[[ | * [[Meridian arc]] | ||
*[[ | * [[Square degree]] | ||
*[[ | * [[Steradian]] | ||
==Notes== | == Notes and references == | ||
{{Reflist}} | {{Reflist|2}} | ||
==External links== | ==External links== | ||
{{Commons category|Degree (angle)}} | {{Commons category|Degree (angle)}} | ||
*[http://www.mathopenref.com/degrees.html Degrees as an angle measure], with interactive animation | * [http://www.mathopenref.com/degrees.html Degrees as an angle measure], with interactive animation | ||
* [http://mathworld.wolfram.com/Degree.html Degree] at [[MathWorld]] | * [http://mathworld.wolfram.com/Degree.html Degree] at [[MathWorld]] | ||
* {{cite web|last=Gray|first=Meghan|title= ° Degree of Angle|url=http://www.sixtysymbols.com/videos/degree.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|coauthors=Merrifield, Michael; Moriarty, Philip|year=2009}} | |||
{{SI units | {{SI units}} | ||
{{DEFAULTSORT:Degree (Angle)}} | {{DEFAULTSORT:Degree (Angle)}} | ||
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[[Category:Imperial units]] | [[Category:Imperial units]] | ||
[[Category:Mathematical constants]] | [[Category:Mathematical constants]] | ||
[[Category:Customary units in the United States | [[Category:Customary units of measurement in the United States]] | ||
Revision as of 00:22, 31 January 2014
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. It is not an SI unit, as the SI unit for angles is radian, but it is mentioned in the SI brochure as an accepted unit.^{[1]} Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians.
History
The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year.^{[2]} Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Persian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers. One of the earliest recorded use of 360 as the total days in a year is found in the Hebrew Bible, Genesis 8-9 as it describes the duration of the Great Flood over 12 months of 30 days each.
Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit and further subdivided the latter into 60 parts following their sexagesimal numeric system.^{[3]} 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.
Aristarchus of Samos and Hipparchos seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically.^{[4]} Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes (DIO 14 ‡2 p. 19 n.24). Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts.
The division of the circle into 360 parts also occurred in ancient India, as evidenced in the Rigveda:^{[5]}
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.
— Dirghatamas , Rigveda 1.164.48
Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24 divisors,^{[6]} making it one of only 7 numbers such that no number less than twice as much has more divisors (sequence A072938 in OEIS).^{[7]} Furthermore, it is divisible by every number from 1 to 10 except 7.^{[8]} This property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention.
Finally, it may be the case that more than one of these factors has come into play. According to that theory, the number is approximately 365 because of the apparent movement of the sun against the celestial sphere and that it was rounded to 360 for some of the mathematical reasons cited above.
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: for example, 40.1875°, with the degree symbol behind the decimals.
Alternatively, the traditional sexagesimal unit subdivisions can be used. 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: for example, 40.1875° = 40° 11′ 15″ . Sometimes single and double quotation marks are used instead: 40° 11' 15" .
If still more accuracy is required, current practice is to use decimal divisions of the second: for example, 40° 11′ 15.4″ . The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today. These subdivisions were denoted{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }} by writing the Roman numeral for the number of sixtieths in superscript: 1^{I} for a "prime" (minute of arc), 1^{II} for a second, 1^{III} for a third, 1^{IV} for a fourth, etc. Hence the modern symbols for the minute and second of arc, and the word "second" also refer to this system. {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }}
Alternative units
{{#invoke:see also|seealso}}
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. Decigrades Template:Frac were used with French artillery sights in World War I.
An angular mil, which is most used in military applications, has at least three specific variants, ranging from Template:Frac to Template:Frac, each approximately equal to one milliradian. However, Template:Frac used by the Russian Army originated in Imperial Russia, where an equilateral chord was divided into tenths to give a circle of 600 units (this may be seen on a dial sight, a device for accurate aiming of artillery, also known as an aiming circle, dating from about 1900, in the St Petersberg Museum of Artillery).
Conversion of some common angles
Units | Values | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Turns | 0 | 1/24 | 1/12 | 1/10 | 1/8 | 1/6 | 1/5 | 1/4 | 1/3 | 2/5 | 1/2 | 3/4 | 1 |
Radians | 0 | ||||||||||||
Degrees | 0° | 15° | 30° | 36° | 45° | 60° | 72° | 90° | 120° | 144° | 180° | 270° | 360° |
Grads | 0^{g} | 40^{g} | 50^{g} | 80^{g} | 100^{g} | 160^{g} | 200^{g} | 300^{g} | 400^{g} |
See also
Notes and references
- ↑ http://www.bipm.org/en/si/si_brochure/chapter4/table6.html
- ↑ Degree, MathWorld
- ↑ J.H. Jeans (1947), The Growth of Physical Science, p.7; Francis Dominic Murnaghan (1946), Analytic Geometry, p.2
- ↑ For more information see D.Rawlins on Aristarchus; and G. J. Toomer, "Hipparchus and Babylonian astronomy."
- ↑ Dirghatamas, Rigveda 1.164.48
- ↑ 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.
- ↑ Divisibility highly composite numbers, Werner Brefeld
- ↑ Contrast this with the relatively unwieldy 2520, which is the least common multiple for every number from 1 to 10.
External links
- Degrees as an angle measure, with interactive animation
- Degree at MathWorld
- Template:Cite web