# Gradian

In trigonometry, the **gradian**, also known as the **gon** (from Ancient Greek: γωνία, romanized: *gōnía*, lit. 'angle'), **grad**,^{[1]} or **grade**,^{[2]} 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).^{[3]}^{[4]}^{[5]} It is equivalent to 1/400 of a turn,^{[6]}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.^{[7]}^{[8]}^{[9]}^{:22}^{[Note 1]}

gon | |
---|---|

Unit of | Angle |

Symbol | ^{g}, gon or grad |

Conversions | |

1 ^{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* was in use for one hundredth of a grad. This 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]}^{[8]}^{[16]} and to a lesser extent in mining^{[17]} and geology.^{[18]}^{[19]}

As of May 2020,^{[update]} 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 German, the unit was formerly also called *Neugrad* (new 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]}^{[8]}^{[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) are 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 Unicode | U+1D4D ᵍ MODIFIER LETTER SMALL G (HTML `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,^{[1]} similarly to a degree sign: 50^{g} = 45°.

## 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 eleven factors of 360 less than or equal to its square root: {2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18}. However, there are only seven for 400: {2, 4, 5, 8, 10, 16, 20}.

## Conversion

Turns | Radians | Degrees | Gradians, or gons |
---|---|---|---|

0 | 0 | 0° | 0^{g} |

1/24 | π/12 | 15° | 16+2/3^{g} |

1/16 | π/8 | 22.5° | 25^{g} |

1/12 | π/6 | 30° | 33+1/3^{g} |

1/10 | π/5 | 36° | 40^{g} |

1/8 | π/4 | 45° | 50^{g} |

1/2π | 1 | c. 57.3° | c. 63.7^{g} |

1/6 | π/3 | 60° | 66+2/3^{g} |

1/5 | 2π/5 | 72° | 80^{g} |

1/4 | π/2 | 90° | 100^{g} |

1/3 | 2π/3 | 120° | 133+1/3^{g} |

2/5 | 4π/5 | 144° | 160^{g} |

1/2 | π | 180° | 200^{g} |

3/4 | 3π/2 | 270° | 300^{g} |

1 | 2π | 360° | 400^{g} |

## Relation to the metre and the SI system of units

In the 18th century, the metre was defined as the 10-millionth part of a quarter meridian. Thus, one grad of arc length along the Earth's surface corresponded to 100 kilometres (assuming a spherical Earth); 1 centigrad of arc equaled 1 kilometre; 0.1 cc (centi-centigrads) of arc equaled 1 metre.^{[25]}

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 10 000 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
- Angular mil (military measurement)
- Harmonic analysis
- Jean-Charles de Borda
- Repeating circle
- Spread (rational trigonometry)
- Steradian (the "square radian")

## Notes

**^**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.^{[10]}The gradations were in one-hundredths of a full revolution.^{[11]}^{[12]}

## References

- ^ a b "List of Geometry and Trigonometry Symbols".
*Math Vault*. 2020-04-17. Retrieved 2020-08-31. **^**Weisstein, Eric W. "Gradian".*mathworld.wolfram.com*. Retrieved 2020-08-31.**^**Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 63, 1998.**^**https://www.nist.gov/pml/nist-guide-si-appendix-b9-factors-units-listed-kind-quantity-or-field-science#ANGLE**^**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.**^**"Gradian".*Art of Problem Solving*. Retrieved 2020-08-31.**^**Balzer, Fritz (1946). Five Place Natural Sine and Tangent Functions in the Centesimal System. Army Map Service, Corps of Engineers, U.S. Army.- ^ 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. **^**Gorini, Catherine A. (2003). The Facts on File Geometry Handbook. Infobase Publishing. ISBN 978-1-4381-0957-2.**^**Cajori, Florian (1899). A History of Physics in Its Elementary Branches: Including the Evolution of Physical Laboratories. Macmillan.The angle through which the torsion-head must be deflected was measured in centesimal divisions of the circle

**^**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]**^**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.

**^**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.**^**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.

- ^ a b Kahmen, Heribert; Faig, Wolfgang (2012). Surveying. De Gruyter. ISBN 9783110845716.
- ^ a b Schofield, Wilfred (2001). Engineering surveying: theory and examination problems for students (5th ed.). Butterworth-Heinemann. ISBN 9780750649872.
- ^ 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. - ^ 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*.**1**. CRC Press. ISBN 978-1-4822-6288-9. - ^ 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. - ^ 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.

**^**"941.202 Einheitenverordnung". Archived from the original on 22 May 2020.- ^ 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
**^**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

**^**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

**^**Cartographie – lecture de carte – Partie H Quelques exemples à retenir. Archived 2 March 2012 at the Wayback Machine**^**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 2017-08-14

## External links

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