Gradian

In trigonometry, the gradianTemplate:Sndalso known as the gon (Template:Etymology), grad, or grade<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>Template:Sndis 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>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite book</ref> It is equivalent to Template:Sfrac of a turn,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Template:Sfrac of a degree, or Template:Sfrac 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">Template:Cite book</ref><ref name="Zimmerman">Template:Cite book</ref><ref name="Gorini 2003">Template:Cite book</ref>Template:Efn

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 minutes and seconds of arc.<ref name="Klein 2012 p. 114">Template:Cite book</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>Template:Citation. On p. 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>Template:Citation</ref>

Gradians (gons) are principally used in surveying (especially in Europe),<ref name="Kahmen and Faig 2012">Template:Cite book</ref><ref name="Zimmerman" /><ref name="Schofield">Template:Cite book</ref> and to a lesser extent in mining<ref name="Sroka">Template:Cite book</ref> and geology.<ref name="Gunzburger et al. 2004">Template:Cite book</ref><ref name="Schmidt and Kühn 2007">Template:Cite book</ref>

The gon (gradian) is a legally recognised unit of measurement in the European Union<ref name="EU units">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>Template:Rp and in Switzerland.<ref name="Switzerland units">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> However, this unit is not part of the International System of Units (SI).<ref name="SI Brochure 9th ed"/><ref name="EU units"/>Template:Rp

History and name

The unit originated in France in connection with the French Revolution as the {{#invoke:Lang|lang}}, 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, Template:Sfrac of a turn), the name gon was later adopted, first in those regions, and later as the international standard.Template:Which In France, it was also called {{#invoke:Lang|lang}}. In German, the unit was formerly also called {{#invoke:Lang|lang}} (new degree) (whereas the standard degree was referred to as {{#invoke:Lang|lang}} (old degree)), likewise {{#invoke:Lang|lang}} in Danish, Swedish and Norwegian (also gradian), and {{#invoke:Lang|lang}} in 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, Template:Sfrac of a turn, or the mathematically more convenient radian, Template:Sfrac of a turn (used in the SI system of units) is generally used instead.

In the Template:Nowrap1990s, most scientific calculators offered the gon (gradian), as well as radians and degrees, for their trigonometric functions.<ref>Template:Citation</ref> In the 2010s, some scientific calculators lack support for gradians.<ref>Template:Citation</ref>

Symbol

Template:Infobox symbol The international standard symbol for this unit is "gon" (see ISO 31-1, Annex B).Template:Needs update Other symbols used in the past include "gr", "grd", and "g", the last sometimes written as a superscript, similarly to a degree sign: 50^g = 45°. A metric prefix is sometimes used, as in "dgon", "cgon", "mgon", denoting respectively 0.1 gon, 0.01 gon, 0.001 gon. Centesimal arc-minutes and centesimal arc-seconds were also denoted with superscripts ^c and ^cc, respectively.

Advantages and disadvantages

Each quadrant is assigned a range of 100 gon, which eases recognition of the four quadrants, as well as arithmetic involving perpendicular or opposite angles.

0°	=	0 gradians
90°	=	100 gradians
180°	=	200 gradians
270°	=	300 gradians
360°	=	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 gon, the direction to one's left is 17 gon, to one's right 217 gon, and behind one 317 gon. A disadvantage is that the common angles of 30° and 60° in geometry must be expressed in fractions (as Template:Sfrac gon and Template:Sfrac gon respectively).

Relation to the metre Template:Anchor

File:Kilometre definition.svg

An early definition of the metre was one ten-millionth of the distance from the North Pole to the equator, measured along a meridian through Paris.

In the 18th century, the metre was defined as the 10-millionth part of a quarter meridian. Thus, 1 gon corresponds to an arc length along the Earth's surface of approximately 100 kilometres; 1 centigon to 1 kilometre; 10 microgons to 1 metre.<ref>Cartographie – lecture de carte – Partie H Quelques exemples à retenir. Template:Webarchive.</ref> (The metre has been redefined with increasing precision since then.)

Relation to the SI system of units Template:Anchor

The gradian is not part of the International System of Units (SI). The EU directive on the units of measurement<ref name="EU units"/>Template:Rp notes that the gradian "does not appear in the lists drawn up by the CGPM, CIPM or BIPM." The most recent, 9th edition of the SI Brochure does not mention the gradian at all.<ref name="SI Brochure 9th ed">Template:SIbrochure9th</ref> The previous edition mentioned it only in the following footnote:<ref name="SI Brochure 8th ed">Template:SIbrochure8th</ref>

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) rad. Thus there are 100 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 Template:Val, 1 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.{{#if:|{{#if:|}}

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See also

• Template:Annotated link
• Template:Annotated link (primarily military use)
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link (the "square radian")

External links

• Ask Dr Math
• Definitions of grade, gon and centigrade on sizes.com
• Dictionary of Units