Gradian

"Grade (angle)" redirects here. For the grade of an inclined surface, see Grade (slope).

"Gon (angle)" redirects here. For n-gon, see Regular polygon.

Not to be confused with Gradient.

In trigonometry, the gradian – also known as the gon (from Ancient Greek γωνία (gōnía) 'angle'), grad, or grade^[1] – 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.^[2][3][4] It is equivalent to ⁠1/400⁠ of a turn,^[5] ⁠9/10⁠ of a degree, or ⁠π/200⁠ of a radian. Measuring angles in gradians is said to employ the centesimal system of angular measurement, initiated as part of metrication and decimalisation efforts.^{[6][7][8][Note 1]}

gon
Compass graded with 400 gon
General information
Unit of	Angle
Symbol	gon, ᵍ, grad
Conversions
1 gon in ...	... is equal to ...
turns	⁠1/400⁠ turn
radians	⁠π/200⁠ rad ≈ 0.0157... rad
milliradians	5π mrad ≈ 15.71... mrad
degrees	⁠9/10⁠°
minutes of arc	54′

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.^[12] The chance of confusion was one reason for the adoption of the term Celsius to replace centigrade as the name of the temperature scale.^[13][14]

Gradians are principally used in surveying (especially in Europe),^[15][7][16] and to a lesser extent in mining^[17] and geology.^[18][19]

The gon is officially a legal unit of measurement in the European Union^[20]: 9 and in Switzerland.^[21] However, the gradian is not part of the International System of Units (SI).^{[22][20]: 9–10}

History and name

The unit originated in France in connection with the French Revolution as the 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, ⁠1/360⁠ of a turn), the name gon was later adopted, first in those regions, and later as the international standard. In France, it was also called grade nouveau. In German, the unit was formerly also called Neugrad (new degree) (whereas the standard degree was referred to as Altgrad (old degree)), likewise nygrad in Danish, Swedish and Norwegian (also gradian), and nýgráða 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,^[15][7][16] mining^[17] and geology.^[18][19] Today, the degree, ⁠1/360⁠ of a turn, or the mathematically more convenient radian, ⁠1/2π⁠ of a turn (used in the SI system of units) is generally used instead.

In the 1970s –1990s, most scientific calculators offered the gon, as well as radians and degrees, for their trigonometric functions.^[23] In the 2010s, some scientific calculators lack support for gradians.^[24]

Symbol

◌ᵍ
Gon
In Unicode	U+1D4D ᵍ MODIFIER LETTER SMALL G
Related
See also	U+00B0 ° DEGREE SIGN

The international standard symbol for this unit today is "gon" (see ISO 31-1). 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.

SI multiples of gon (gon)

Submultiples			Multiples
Value	SI symbol	Name	Value	SI symbol	Name
10−1 gon	dgon	decigon	101 gon	dagon	decagon
10−2 gon	cgon	centigon	102 gon	hgon	hectogon
10−3 gon	mgon	milligon	103 gon	kgon	kilogon
10−6 gon	μgon	microgon	106 gon	Mgon	megagon
10−9 gon	ngon	nanogon	109 gon	Ggon	gigagon
10−12 gon	pgon	picogon	1012 gon	Tgon	teragon
10−15 gon	fgon	femtogon	1015 gon	Pgon	petagon
10−18 gon	agon	attogon	1018 gon	Egon	exagon
10−21 gon	zgon	zeptogon	1021 gon	Zgon	zettagon
10−24 gon	ygon	yoctogon	1024 gon	Ygon	yottagon
10−27 gon	rgon	rontogon	1027 gon	Rgon	ronnagon
10−30 gon	qgon	quectogon	1030 gon	Qgon	quettagon

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 ⁠33+1/3⁠ gon and ⁠66+2/3⁠ gon respectively).

Conversion

Conversion of common angles

Turns	Radians	Degrees	Gradians
0 turn	0 rad	0°	0g
⁠1/72⁠ turn	⁠π/36⁠ or ⁠𝜏/72⁠ rad	5°	⁠5+5/9⁠g
⁠1/24⁠ turn	⁠π/12⁠ or ⁠𝜏/24⁠ rad	15°	⁠16+2/3⁠g
⁠1/16⁠ turn	⁠π/8⁠ or ⁠𝜏/16⁠ rad	22.5°	25g
⁠1/12⁠ turn	⁠π/6⁠ or ⁠𝜏/12⁠ rad	30°	⁠33+1/3⁠g
⁠1/10⁠ turn	⁠π/5⁠ or ⁠𝜏/10⁠ rad	36°	40g
⁠1/8⁠ turn	⁠π/4⁠ or ⁠𝜏/8⁠ rad	45°	50g
⁠1/2π or 𝜏⁠ turn	1 rad	approx. 57.3°	approx. 63.7g
⁠1/6⁠ turn	⁠π/3⁠ or ⁠𝜏/6⁠ rad	60°	⁠66+2/3⁠g
⁠1/5⁠ turn	⁠2π or 𝜏/5⁠ rad	72°	80g
⁠1/4⁠ turn	⁠π/2⁠ or ⁠𝜏/4⁠ rad	90°	100g
⁠1/3⁠ turn	⁠2π or 𝜏/3⁠ rad	120°	⁠133+1/3⁠g
⁠2/5⁠ turn	⁠4π or 2𝜏 or α/5⁠ rad	144°	160g
⁠1/2⁠ turn	π or ⁠𝜏/2⁠ rad	180°	200g
⁠3/4⁠ turn	⁠3π or ρ/2⁠ or ⁠3𝜏/4⁠ rad	270°	300g
1 turn	𝜏 or 2π rad	360°	400g

Relation to the metre

Further information: History of the metre § Meridional definition

See also: Arc length § Great circles on Earth, and Nautical mile § Similar definitions

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.^[25] (The metre has been redefined with increasing precision since then.)

Relation to the SI system of units

The gradian is not part of the International System of Units (SI). The EU directive on the units of measurement^{[20]: 9–10} 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.^[22] The previous edition mentioned it only in the following footnote:^[26]

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 10000 km, 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.

See also

• Angular frequency – Rate of change of angle
• Milliradian – Angular measurement, thousandth of a radian (primarily military use)
• Harmonic analysis – Study of superpositions in mathematics
• Jean-Charles de Borda – French scientist and Navy officer (1733–1799)
• Repeating circle – type of angular measurement instrument, notably used for navigationPages displaying wikidata descriptions as a fallback
• Spread (rational trigonometry) – 2005 book reformulating plane geometryPages displaying short descriptions of redirect targets
• Steradian – SI derived unit of solid angle (the "square radian")

Notes

1. 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 Ref.^[9] The gradations were in one-hundredths of a full revolution.^[10][11]

References

1. Weisstein, Eric W. "Gradian". mathworld.wolfram.com. Retrieved 2020-08-31.
2. Harris, J. W.; Stocker, H. (1998). Handbook of Mathematics and Computational Science. New York: Springer-Verlag. p. 63.
3. "NIST Guide to the SI, Appendix B.9: Factors for units listed by kind of quantity or field of science". nist.gov. NIST. Archived from the original on 2017-04-17.
4. Patrick Bouron (2005). Cartographie: Lecture de Carte (PDF). Institut Géographique National. p. 12. Archived from the original (PDF) on 2010-04-15. Retrieved 2011-07-07.
5. "Gradian". Art of Problem Solving. Retrieved 2020-08-31.
6. Balzer, Fritz (1946). Five Place Natural Sine and Tangent Functions in the Centesimal System. Army Map Service, Corps of Engineers, U.S. Army.
7. Zimmerman, Edward G. (1995). "6. Angle Measurement: Transits and Theodolites". In Minnick, Roy; Brinker, Russell Charles (eds.). The surveying handbook (2nd ed.). Chapman & Hall. ISBN 041298511X.
8. Gorini, Catherine A. (2003). The Facts on File Geometry Handbook. Infobase Publishing. p. 22. ISBN 978-1-4381-0957-2.
9. Cajori, Florian (1899). A History of Physics in Its Elementary Branches: Including the Evolution of Physical Laboratories. Macmillan. ISBN 9781548494957. The angle through which the torsion-head must be deflected was measured in centesimal divisions of the circle
10. Ohm, Georg Simon (1826). "Bestimmung des Gesetzes, nach welchem Metalle die Contactelektricität leiten, nebst einem Entwurfe zur Theorie des Voltaischen Apparates und des Schweiggerschen Multiplikators" (PDF). Journal für Chemie und Physik. 46: 137–166. Archived from the original (PDF) on 23 May 2020. German: 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]
11. Keithley, Joseph F. (1999). The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s. John Wiley & Sons. ISBN 978-0-7803-1193-0. It hung on a ribbon torsion element with a knob on top, graduated in 100 parts.
12. Klein, H.A. (2012). The Science of Measurement: A Historical Survey. Dover Books on Mathematics. Dover Publications. p. 114. ISBN 978-0-486-14497-9. Retrieved 2022-01-02.
13. Frasier, E. Lewis (February 1974), "Improving an imperfect metric system", Bulletin of the Atomic Scientists, 30 (2): 9–44, Bibcode:1974BuAtS..30b...9F, doi:10.1080/00963402.1974.11458078. 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.
14. Mahaffey, Charles T. (1976), "Metrication problems in the construction codes and standards sector", Final Report National Bureau of Standards, NBS Technical Note 915, U.S. Department of Commerce, National Bureau of Commerce, Institute for Applied Technology, Center for Building Technology, Bibcode:1976nbs..reptU....M, The term "Celsius" was adopted instead of the more familiar "centigrade" because in France the word centigrade has customarily been applied to angles.
15. Kahmen, Heribert; Faig, Wolfgang (2012). Surveying. De Gruyter. ISBN 9783110845716.
16. Schofield, Wilfred (2001). Engineering surveying: theory and examination problems for students (5th ed.). Butterworth-Heinemann. ISBN 9780750649872.
17. Sroka, Anton (2006). "Contribution to the prediction of ground surface movements caused by a rising water level in a flooded mine". In Sobczyk, Eugeniusz; Kicki, Jerzy (eds.). International Mining Forum 2006, New Technological Solutions in Underground Mining: Proceedings of the 7th International Mining Forum, Cracow - Szczyrk - Wieliczka, Poland, February 2006. CRC Press. ISBN 9780415889391.
18. Gunzburger, Yann; Merrien-Soukatchoff, Véronique; Senfaute, Gloria; Piguet, Jack-Pierre; Guglielmi, Yves (2004). "Field investigations, monitoring and modeling in the identification of rock fall causes". In Lacerda, W.; Ehrlich, Mauricio; Fontoura, S. A. B.; Sayão, A. S. F. (eds.). 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. Vol. 1. CRC Press. ISBN 978-1-4822-6288-9.
19. Schmidt, Dietmar; Kühn, Friedrich (2007). "3. Remote sensing: 3.1 Aerial Photography". In Knödel, Klaus; Lange, Gerhard; Voigt, Hans-Jürgen (eds.). Environmental Geology: Handbook of Field Methods and Case Studies. Springer Science & Business Media. ISBN 978-3-540-74671-3.
20. "Directive 80/181/EEC". 27 May 2009. Archived from the original on 22 May 2020. On the approximation of the laws of the Member States relating to units of measurement and on the repeal of Directive 71/354/EEC.
21. "941.202 Einheitenverordnung" (in German). Archived from the original on 22 May 2020.
22. The International System of Units (PDF) (9th ed.), International Bureau of Weights and Measures, Dec 2022, ISBN 978-92-822-2272-0
23. Maloney, Timothy J. (1992), Electricity: Fundamental Concepts and Applications, Delmar Publishers, p. 453, ISBN 9780827346758, On most scientific calculators, this [the unit for angles] is set by the DRG key
24. Cooke, Heather (2007), Mathematics for Primary and Early Years: Developing Subject Knowledge, SAGE, p. 53, ISBN 9781847876287, Scientific calculators commonly have two modes for working with angles – degrees and radians
25. Cartographie – lecture de carte – Partie H Quelques exemples à retenir. Archived 2 March 2012 at the Wayback Machine.
26. International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16

External links

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