# Gradian

In trigonometry, the gradian, also known as the gon (from Ancient Greek: γωνία, romanizedgōnía, lit.'angle'), grad, or grade,[1] is a unit of measurement of an angle, defined as one hundredth of the right angle; in other words, there are 100 gradians in 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]

Compass graded with 400 gon.
gon
Unit ofAngle
Symbolg, gon or grad
Conversions
g 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 grad; 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]

As of May 2020, the gon is officially a legal unit of measurement in the European Union[20]: 9  and in Switzerland.[21]

The gradian is not part of the International System of Units (SI).[22][20]: 9–10

## History and name

The unit originated in connection with the French Revolution in France 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] The French artillery[who?] has used the grad for decades.[citation needed] 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 grad, 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 UnicodeU+1D4D MODIFIER LETTER SMALL G
Related
See alsoU+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: 50g = 45°. A metric prefix sometimes is used, as in "dgon", "cgon", "mgon", 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 grad, the direction to one's left is 17 grad, to one's right 217 grad, and behind one 317 grad. A disadvantage is that the common angles of 30° and 60° in geometry must be expressed in fractions (as 33+1/3 grad and 66+2/3 grad, respectively).

Similarly, in one hour (1/24 day), Earth rotates by 15° or 16+2/3 gon (see also decimal time). These observations are a consequence of the fact that the number 360 has more divisors than the number 400 does; notably, 360 is divisible by 3, while 400 is not. There are twelve factors of 360 less than or equal to its square root: {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18}. However, there are only eight for 400: {1, 2, 4, 5, 8, 10, 16, 20}.

## Conversion

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

## Relation to the metre

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, one grad corresponds to an arc length along the Earth's surface of approximately 100 kilometres; 1 centigrad to 1 kilometre; 10 micrograd to 1 metre.[25]

## 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 a footnote, which said the following:[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.

## 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. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 63, 1998.
3. ^ "NIST Guide to the SI, Appendix B.9: Factors for units listed by kind of quantity or field of science | NIST". www.nist.gov. 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. ^ a b c 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. ^ a b Kahmen, Heribert; Faig, Wolfgang (2012). Surveying. De Gruyter. ISBN 9783110845716.
16. ^ a b Schofield, Wilfred (2001). Engineering surveying: theory and examination problems for students (5th ed.). Butterworth-Heinemann. ISBN 9780750649872.
17. ^ a b 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. ^ a b 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. ^ a b 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. ^ a b c "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". Archived from the original on 22 May 2020.
22. ^ a b International Bureau of Weights and Measures (2019-05-20), SI Brochure: The International System of Units (SI) (PDF) (9th ed.), ISBN 978-92-822-2272-0`{{citation}}`: CS1 maint: url-status (link)
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. ^
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