View source for Gradian

{{short description|Unit of measurement of an angle, equal to 1/400th of a circle}}
{{redirect|Grade (angle)|the grade of an inclined surface|Grade (slope)}}
{{Redirect|Gon (angle)|n-gon|Regular polygon}}
{{distinguish|Gradient}}
{{Infobox unit
| name       = gon
| othernames = gradian, grade, grad
| image      = Boussole en grades table conversion (cropped).jpg
| caption    = [[Compass]] graded with 400 gon
| standard   = 
| quantity   = [[Angle]]
| symbol     = gon
| symbol2    = &#x1D4D;
| symbol3    = grad
| units1     = [[Turn (angle)|turn]]s
| inunits1   = {{sfrac|1|400}} turn
| units2     = [[radian]]s
| inunits2   = {{sfrac|{{pi}}|200}}&nbsp;rad <br />≈ 0.0157...&nbsp;rad
| units3     = [[milliradian]]s
| inunits3   = 5{{pi}}&nbsp;mrad <br />≈ 15.71...&nbsp;mrad
| units4     = [[Degree (angle)|degree]]s
| inunits4   = 0.9°
| units5     = [[minute and second of arc|minutes of arc]]
| inunits5   = 54′
}}

In [[trigonometry]], the '''gradian'''{{snd}}also known as the '''gon''' ({{etymology|grc|''{{Wikt-lang|grc|γωνία}}'' ({{grc-transl|γωνία}})|angle}}), '''grad''', or '''grade'''<ref>{{cite web |last=Weisstein|first=Eric W. |title=Gradian |url=https://mathworld.wolfram.com/Gradian.html |access-date=2020-08-31|website=mathworld.wolfram.com |language=en}}</ref>{{snd}}is a [[unit of measurement]] of an [[angle]], defined as one-hundredth of the [[right angle]]; in other words, 100 gradians is equal to 90 degrees.<ref name=handbookOfMath>{{cite book |last1=Harris|first1=J. W. |last2=Stocker|first2=H. |title=Handbook of Mathematics and Computational Science |location=New York |publisher=[[Springer-Verlag]] |page=63 |year=1998}}</ref><ref>{{cite web |url=https://nist.gov/pml/nist-guide-si-appendix-b9-factors-units-listed-kind-quantity-or-field-science#ANGLE |title=NIST Guide to the SI, Appendix B.9: Factors for units listed by kind of quantity or field of science |publisher=[[National Institute of Standards and Technology|NIST]] |website=nist.gov |url-status=dead |archive-url=https://web.archive.org/web/20170417071705/https://www.nist.gov/pml/nist-guide-si-appendix-b9-factors-units-listed-kind-quantity-or-field-science |archive-date=2017-04-17}}</ref><ref>{{cite book|url=http://webdav-noauth.unit-c.fr/files/perso/pbarbier/cours_unit/Elements_de_base_de_la_cartographie.pdf|page=12|date=2005|publisher=Institut Géographique National|author=Patrick Bouron|access-date=2011-07-07|title=Cartographie: Lecture de Carte|url-status=dead|archive-url=https://web.archive.org/web/20100415034329/http://webdav-noauth.unit-c.fr/files/perso/pbarbier/cours_unit/Elements_de_base_de_la_cartographie.pdf|archive-date=2010-04-15}}</ref> It is equivalent to {{sfrac|1|400}} of a [[turn (geometry)|turn]],<ref>{{Cite web|title=Gradian|url=https://artofproblemsolving.com/wiki/index.php/Gradian|access-date=2020-08-31|website=Art of Problem Solving}}</ref> {{sfrac|9|10}} of a [[degree (angle)|degree]], or {{sfrac|{{pi}}|200}} of a [[radian]]. Measuring angles in gradians (gons) is said to employ the ''centesimal'' system of angular measurement, initiated as part of [[metrication]] and [[decimalisation]] efforts.<ref name="Balzer 1946">{{cite book |last1=Balzer |first1=Fritz |title=Five Place Natural Sine and Tangent Functions in the Centesimal System |date=1946 |publisher=Army Map Service, Corps of Engineers, U.S. Army. |url=https://books.google.com/books?id=2IxG7M9Yw0UC&q=centesimal&pg=PP9 |language=en}}</ref><ref name="Zimmerman">{{cite book |last1=Zimmerman |first1=Edward G.|title=The surveying handbook |publisher=Chapman & Hall|editor-first1=Roy |editor-last1=Minnick| editor-first2=Russell Charles |editor-last2=Brinker | chapter= 6. Angle Measurement: Transits and Theodolites |isbn=041298511X |edition=2nd|year=1995|chapter-url=https://books.google.com/books?id=2gB7w9XlNJAC&q=centesimal&pg=PA80}}</ref><ref name="Gorini 2003">{{cite book |last1=Gorini |first1=Catherine A. |title=The Facts on File Geometry Handbook |date=2003 |publisher=Infobase Publishing |isbn=978-1-4381-0957-2 |url=https://books.google.com/books?id=PlYCcvgLJxYC |language=en |page=[https://books.google.com/books?id=PlYCcvgLJxYC&q=centesimal&pg=PA22 22]}}</ref>{{efn|On rare occasions, ''centesimal'' refers to the division of the full angle (360°) into hundred parts. One example is the description of the gradations on [[Georg Ohm]]'s  torsion balance in a history of physics book from 1899.<ref name="Cajori 1899">{{cite book |last1=Cajori |first1=Florian |title=A History of Physics in Its Elementary Branches: Including the Evolution of Physical Laboratories |date=1899 |publisher=Macmillan |isbn=9781548494957 |url=https://books.google.com/books?id=7Y5KAAAAMAAJ&dq=%22history%22%20centesimal%20angle&pg=PA229 |language=en|quote="The angle through which the torsion-head must be deflected was measured in centesimal divisions of the circle"}}</ref> The gradations were in one-hundredths of a full revolution.<ref name="Ohm 1826">{{cite journal|first=Georg Simon|last=Ohm|title=Bestimmung des Gesetzes, nach welchem Metalle die Contactelektricität leiten, nebst einem Entwurfe zur Theorie des Voltaischen Apparates und des Schweiggerschen Multiplikators|journal=Journal für Chemie und Physik|year=1826|volume=46|pages=137–166|url=http://www2.ohm-hochschule.de/bib/textarchiv/Ohm.Bestimmung_des_Gesetzes.pdf|archive-url=https://web.archive.org/web/20200523161946/http://www2.ohm-hochschule.de/bib/textarchiv/Ohm.Bestimmung_des_Gesetzes.pdf|archive-date=23 May 2020|quote={{langx|de|wurde die Größe der Drehung oben an der Drehwage in Hunderttheilen einer ganzen Umdrehung abgelesen}} (p. 147) [the amount of rotation at the top of the torsion balance was read in hundred parts of an entire revolution]}}</ref><ref name="Keithley 1999">{{cite book |last1=Keithley |first1=Joseph F. |title=The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s |date=1999 |publisher=John Wiley & Sons |isbn=978-0-7803-1193-0 |url=https://www.google.com/search?tbm=bks&q=%22It+hung+on+a+ribbon+torsion+element+with+a+knob+on+top%2C+graduated+in+100+parts.%22 |language=en|quote=It hung on a ribbon torsion element with a knob on top, graduated in 100 parts.}}</ref>}}

In continental [[Europe]], the French word '''''centigrade''''', also known as ''centesimal minute of arc'', was in use for one hundredth of a grade; similarly, the ''centesimal second of arc'' was defined as one hundredth of a centesimal arc-minute, analogous to [[decimal time]] and the [[sexagesimal degrees|sexagesimal]] [[minutes and seconds of arc]].<ref name="Klein 2012 p. 114">{{cite book | last=Klein | first=H.A. | title=The Science of Measurement: A Historical Survey | publisher=Dover Publications | series=Dover Books on Mathematics | year=2012 | isbn=978-0-486-14497-9 | url=https://books.google.com/books?id=CrmuSiCFyikC&pg=PA114 | access-date=2022-01-02 | page=114}}</ref> The chance of confusion was one reason for the adoption of the term ''[[Celsius]]'' to replace ''centigrade'' as the name of the temperature scale.<ref>{{citation|title=Improving an imperfect metric system|first=E. Lewis|last=Frasier|journal=Bulletin of the Atomic Scientists|date=February 1974|volume=30|issue=2|pages=9–44|doi=10.1080/00963402.1974.11458078|bibcode=1974BuAtS..30b...9F}}. On [https://books.google.com/books?id=IwwAAAAAMBAJ&pg=PA42 p.&nbsp;42] Frasier argues for using grads instead of radians as a standard unit of angle, but for renaming grads to "radials" instead of renaming the temperature scale.</ref><ref>{{citation|title=Metrication problems in the construction codes and standards sector|series=NBS Technical Note 915|first=Charles T.|last=Mahaffey|journal=Final Report National Bureau of Standards|publisher=U.S. Department of Commerce, National Bureau of Commerce, Institute for Applied Technology, Center for Building Technology|year=1976|bibcode=1976nbs..reptU....M|url=https://archive.org/stream/metricationprobl915maha|quote=The term "Celsius" was adopted instead of the more familiar "centigrade" because in France the word centigrade has customarily been applied to angles.}}</ref>

Gradians (gons) are principally used in [[surveying]] (especially in Europe),<ref  name="Kahmen and Faig 2012">{{cite book|first1= Heribert |last1=Kahmen|first2=Wolfgang|last2= Faig |title=Surveying |isbn=9783110845716|publisher=De Gruyter|year=2012|url=https://books.google.com/books?id=7nMiAAAAQBAJ&dq=%22gon%22%20%20%22surveying%22%20%22Europe%22&pg=PA58}}</ref><ref name="Zimmerman" /><ref name="Schofield">{{cite book |first=Wilfred |last=Schofield|title=Engineering surveying: theory and examination problems for students |publisher=Butterworth-Heinemann |isbn=9780750649872 |edition=5th|year=2001|url=https://books.google.com/books?id=1wYBMUYV0b0C&q=gon&pg=PA23}}</ref> 
<!--
in [[stepper motor]]s<ref>{{cite book |last1=Benaron |first1=David A. |last2=Ho |first2=David C. |last3=Spilman |first3=Stanley |last4=Van Houten |first4=John P. |last5=Stevenson |first5=David K. |chapter=Tomographic Time-of-Flight Optical Imaging Device|editor-last1=Hogan |editor-first1=Michael C. |editor-last2=Mathieu-Costello |editor-first2=Odile |editor-last3=Poole |editor-first3=David C. |editor-last4=Wagner |editor-first4=Peter D.  |title=Oxygen Transport to Tissue XVI |series=Advances in Experimental Medicine and Biology |date=1994 |volume=361 |pages=207–214 |doi=10.1007/978-1-4615-1875-4_26 |chapter-url=https://books.google.com/books?id=2HnrBwAAQBAJ&dq=%22we%20found%20that%20the%20convenience%20of%20using%20one%20gradian%20per%20step%20facilitated%20use%20of%20faster%20integer%20calculations%20in%20our%20processing%20algorithms%2C%22&pg=PA209|publisher=Springer US |pmid=7597945 |isbn=978-1-4613-5763-6 |language=en|quote="we found that the convenience of using one gradian per step facilitated use of faster integer calculations in our processing algorithms}}</ref>
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aircraft navigation,<ref name="Gorini 2003">{{cite book |last1=Gorini |first1=Catherine A. |title=The Facts on File Geometry Handbook |date=2003 |publisher=Infobase Publishing |isbn=978-1-4381-0957-2 |url=https://books.google.com/books?id=PlYCcvgLJxYC |language=en}}</ref><sup>:[https://books.google.com/books?id=PlYCcvgLJxYC&q=aircraft%20navigation&pg=PA73 73]</sup> 
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and to a lesser extent in [[mining]]<ref name="Sroka">{{cite book |title=International Mining Forum 2006, New Technological Solutions in Underground Mining: Proceedings of the 7th International Mining Forum, Cracow - Szczyrk - Wieliczka, Poland, February 2006  |isbn=9780415889391|year=2006|editor-first1=Eugeniusz |editor-last1=Sobczyk|editor-first2= Jerzy|editor-last2=Kicki|publisher=[[CRC Press]]|chapter=Contribution to the prediction of ground surface movements caused by a rising water level in a flooded mine|author-first=Anton|author-last=Sroka|chapter-url=https://books.google.com/books?id=-smnDwAAQBAJ&q=gon&pg=PA68}}</ref> and [[geology]].<ref name="Gunzburger et al. 2004">{{cite book |editor-last1=Lacerda |editor-first1=W. |editor-last2=Ehrlich |editor-first2=Mauricio |editor-last3=Fontoura |editor-first3=S. A. B. |editor-last4=Sayão |editor-first4=A. S. F. |title=Landslides: Evaluation & Stabilization/Glissement de Terrain: Evaluation et Stabilisation, Set of 2 Volumes: Proceedings of the Ninth International Symposium on Landslides, June 28 -July 2, 2004 Rio de Janeiro, Brazil |date=2004 |volume=1|publisher=CRC Press |isbn=978-1-4822-6288-9 |author-last1=Gunzburger|author-first1=Yann|author-last2=Merrien-Soukatchoff|author-first2=Véronique|author-last3=Senfaute|author-first3=Gloria|author-last4=Piguet|author-first4=Jack-Pierre|author-last5=Guglielmi|author-first5=Yves|chapter=Field investigations, monitoring and modeling in the identification of rock fall causes|chapter-url=https://books.google.com/books?id=DLPNBQAAQBAJ&dq=mining%20measurement%20%22gon%22%20-%22data%20mining%22&pg=PA561}}</ref><ref name="Schmidt and Kühn 2007">{{cite book |editor-last1=Knödel |editor-first1=Klaus |editor-last2=Lange |editor-first2=Gerhard |editor-last3=Voigt |editor-first3=Hans-Jürgen |title=Environmental Geology: Handbook of Field Methods and Case Studies |date=2007 |publisher=[[Springer Science & Business Media]] |isbn=978-3-540-74671-3 |author-first1= Dietmar|author-last1=Schmidt |author-first2=Friedrich |author-last2=Kühn|chapter=3. Remote sensing: 3.1 Aerial Photography|chapter-url=https://books.google.com/books?id=QF5_Xvhm8KQC&dq=geology%20%22gon%22&pg=PA28}}</ref>

The gon (gradian) is a legally recognised unit of measurement in the [[European Union]]<ref name="EU units">{{Cite web |url=https://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:01980L0181-20090527&from=EN |title=Directive 80/181/EEC |date=27 May 2009 |quote=On the approximation of the laws of the Member States relating to units of measurement and on the repeal of Directive 71/354/EEC. |archive-url=https://web.archive.org/web/20200522202337/https://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX%3A01980L0181-20090527&from=EN |archive-date=22 May 2020}}</ref>{{rp|9}} and in [[Switzerland]].<ref name="Switzerland units">{{cite web |url=https://www.admin.ch/opc/de/classified-compilation/19940345/ |title=941.202 Einheitenverordnung |language=de |archive-url=https://web.archive.org/web/20200522202750/https://www.admin.ch/opc/de/classified-compilation/19940345/ |archive-date=22 May 2020}}</ref> However, this unit is not part of the [[International System of Units]] (SI).<ref name="SI Brochure 9th ed"/><ref name="EU units"/>{{rp|9–10}}

== History and name==
The unit originated in [[France]] in connection with the [[French Revolution]] as the {{lang|fr|grade}}, along with the [[metric system]], hence it is occasionally referred to as a ''metric degree''. Due to confusion with the existing term ''grad(e)'' in some northern European countries (meaning a standard degree, {{sfrac|1|360}} of a turn), the name ''gon'' was later adopted, first in those regions, and later as the international standard.{{which|date=December 2024}} In France, it was also called {{lang|fr|grade nouveau}}. In [[German language|German]], the unit was formerly also called {{lang|de|Neugrad}} (new degree) (whereas the standard degree was referred to as {{lang|de|Altgrad}} (old degree)), likewise {{lang|da|nygrad}} in [[Danish language|Danish]], [[Swedish language|Swedish]] and [[Norwegian language|Norwegian]] (also ''gradian''), and {{lang|is|nýgráða}} in [[Icelandic language|Icelandic]].

Although attempts at a general introduction were made, the unit was only adopted in some countries, and for specialised areas such as [[surveying]],<ref  name="Kahmen and Faig 2012"/><ref name="Zimmerman" /><ref name="Schofield"/> [[mining]]<ref name="Sroka"/> and [[geology]].<ref name="Gunzburger et al. 2004"/><ref name="Schmidt and Kühn 2007"/> Today, the degree, {{sfrac|1|360}} of a [[Turn (geometry)|turn]], or the mathematically more convenient [[radian]], {{sfrac|1|2{{pi}}}} of a turn (used in the [[SI]] system of units) is generally used instead.

In the {{nowrap|1970s{{hsp}}{{ndash}}}}1990s, most [[scientific calculator]]s offered the gon (gradian), as well as radians and degrees, for their [[trigonometric functions]].<ref>{{citation|title=Electricity: Fundamental Concepts and Applications|first=Timothy J.|last=Maloney|publisher=Delmar Publishers|year=1992|isbn=9780827346758|page=[https://archive.org/details/electricityfunda0000malo/page/453 453]|quote=On most scientific calculators, this [the unit for angles] is set by the DRG key|url=https://archive.org/details/electricityfunda0000malo/page/453}}</ref> In the 2010s, some scientific calculators<!-- from Texas Instruments and at least one from Hewlett-Packard --> lack support for gradians.<ref>{{citation|title=Mathematics for Primary and Early Years: Developing Subject Knowledge|first=Heather|last=Cooke|publisher=SAGE|year=2007|isbn=9781847876287|page=53|url=https://books.google.com/books?id=C3t4B3stB0UC&pg=PA53|quote=Scientific calculators commonly have two modes for working with angles – degrees and radians}}</ref>

=== Symbol ===
{{infobox symbol
|name= Gon
|mark = ◌&#x1D4D;   
|unicode={{unichar|1D4D|modifier letter small g|nlink=modifier letter}}
|see also={{unichar|00B0|nlink=Degree sign}}
}}
The international standard symbol for this unit is "gon" (see [[ISO 31-1]], Annex B).{{needs update|reason=ISO 31-1 has been superseded twice, so this information may be obsolete.|date=December 2024}} Other symbols used in the past include "gr", "grd", and "g", the last sometimes written as a superscript, similarly to a degree sign: 50<sup>g</sup> = 45°.
A [[metric prefix]] is sometimes used, as in "dgon", "cgon", "mgon", denoting respectively 0.1&nbsp;gon, 0.01&nbsp;gon, 0.001&nbsp;gon.
Centesimal arc-minutes and centesimal arc-seconds were also denoted with superscripts <sup>c</sup> and <sup>cc</sup>, respectively.

{{SI multiples
| unit=gon
| symbol=gon
}}

== Advantages and disadvantages ==

Each quadrant is assigned a range of 100&nbsp;gon, which eases recognition of the four quadrants, as well as arithmetic involving perpendicular or opposite angles.
: {|
|-
|align="right"|   0° ||align="center"| = ||align="right"|   0 gradians
|-
|align="right"|  90° ||align="center"| = ||align="right"| 100 gradians
|-
|align="right"| 180° ||align="center"| = ||align="right"| 200 gradians
|-
|align="right"| 270° ||align="center"| = ||align="right"| 300 gradians
|-
|align="right"| 360° ||align="center"| = ||align="right"| 400 gradians
|}

One advantage of this unit is that right angles to a given angle are easily determined. If one is sighting down a compass course of 117&nbsp;gon, the direction to one's left is 17&nbsp;gon, to one's right 217&nbsp;gon, and behind one 317&nbsp;gon. A disadvantage is that the common angles of 30° and 60° in geometry must be expressed in fractions (as {{sfrac|33|1|3}}&nbsp;gon and {{sfrac|66|2|3}}&nbsp;gon respectively).

== Conversion ==

{{Table of angles}}

== Relation to the metre <span class="anchor" id="Metre"></span> ==
{{further|History of the metre#Meridional definition}}
{{see also|Arc length#Great circles on Earth|Nautical mile#Similar definitions}}
[[File:Kilometre definition.svg|thumb|An early definition of the [[metre]] was one ten-millionth of the distance from the [[North Pole]] to the [[equator]], measured along a [[meridian (geography)|meridian]] through [[Paris]].]]

In the 18th century, the [[metre]] was defined as the 10-millionth part of a [[quarter meridian]].
Thus, 1&nbsp;gon corresponds to an [[arc length]] along the Earth's surface of approximately 100&nbsp;kilometres; 1&nbsp;centigon to 1&nbsp;kilometre; 10&nbsp;microgons to 1&nbsp;metre.<ref>[http://webdav-noauth.unit-c.fr/files/perso/pbarbier/cours_unit/Elements_de_base_de_la_cartographie.pdf Cartographie – lecture de carte – Partie H Quelques exemples à retenir]. {{webarchive|url=https://web.archive.org/web/20120302111251/http://webdav-noauth.unit-c.fr/files/perso/pbarbier/cours_unit/Elements_de_base_de_la_cartographie.pdf|date=2 March 2012}}.</ref> (The [[metre#History of definition|metre has been redefined]] with increasing precision since then.)

== Relation to the SI system of units <span class="anchor" id="SI"></span> ==
The gradian is ''not'' part of the [[International System of Units]] (SI). The EU directive on the units of measurement<ref name="EU units"/>{{rp|9–10}} notes that the gradian "does not appear in the lists drawn up by the [[General Conference on Weights and Measures|CGPM]], [[General Conference on Weights and Measures#CIPM|CIPM]] or [[International Bureau of Weights and Measures|BIPM]]." The most recent, 9th edition of the [[International System of Units#SI Brochure|SI Brochure]] does not mention the gradian at all.<ref name="SI Brochure 9th ed">{{SIbrochure9th}}</ref> The previous edition mentioned it only in the following footnote:<ref name="SI Brochure 8th ed">{{SIbrochure8th}}</ref> {{blockquote|The gon (or grad, where grad is an alternative name for the gon) is an alternative unit of plane angle to the degree, defined as (π/200)&nbsp;rad. Thus there are 100&nbsp;gon in a right angle. The potential value of the gon in navigation is that because the distance from the pole to the equator of the Earth is approximately {{val|10000|u=km}}, 1&nbsp;km on the surface of the Earth subtends an angle of one centigon at the centre of the Earth. However the gon is rarely used.}}

== See also ==
* {{Annotated link |Angular frequency}}
* {{Annotated link |Milliradian}} (primarily military use)
* {{Annotated link |Harmonic analysis}}
* {{Annotated link |Jean-Charles de Borda}}
* {{Annotated link |Repeating circle}}
* {{Annotated link |Spread (rational trigonometry)}}
* {{Annotated link |Steradian}} (the "square radian")

== Notes ==
{{notelist}}

== References ==
{{Reflist}}

== External links ==
* [http://mathforum.org/library/drmath/view/55451.html Ask Dr Math]
* Definitions of [http://www.sizes.com/units/grade.htm grade], [http://www.sizes.com/units/gon.htm gon] and [http://www.sizes.com/units/temperature_centigrade.htm centigrade] on sizes.com
* [http://www.cleavebooks.co.uk/dictunit/notesa.htm Dictionary of Units]

[[Category:Units of plane angle]]
[[Category:Decimalisation]]
[[Category:Metrication]]
[[Category:Non-SI metric units]]