# Gradian

The **gradian** 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).[1][2][3][4]
It is equivalent to 1/400 of a turn, 9/10 of a degree, or π/200 of a radian.

Gon | |
---|---|

Unit of | Angle |

Symbol | ᵍ or gon |

Conversions | |

1 ᵍ in ... | ... is equal to ... |

turns | 1/400 turn |

radians | π/200 rad |

degrees | 9/10° |

minutes of arc | 54′ |

It is also known as **gon** (from Greek γωνία/*gōnía* for angle), **grad**, or **grade**. In continental Europe, the French term *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.[5][6]

## History

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, later as the international standard. In German, the unit was formerly also called *Neugrad* (new degree), likewise *nygrad* in Swedish, Danish 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, mining and geology. The French artillery has used the grad for decades. 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 and 1980s most scientific calculators offered the grad as well as radians and degrees for their trigonometric functions.[7] In the 2010s, some scientific calculators lack support for gradians.[8]

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°.

## 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 (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}.

In the 18th century, the metre was defined as the forty-millionth part of a meridian. Thus, one grad of arc along the Earth's surface corresponded to 100 kilometres of distance at the equator; 1 centigrad of arc equaled 1 kilometre; 0.1 cc (centi-centigrads) of arc equaled 1 metre.[9]

## Use in surveying

The gradian is sometimes used in surveying in some parts of the world.[10] Subdivisions of the gradian used in surveying can be referred to as c (short for centigrad), sometimes referred to as a *new minute*, and cc (effectively centi-centigrad), sometimes referred to as a "new second," where 1 c = 0.01 grad and 1 cc = 0.0001 grad.

## Conversion

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

0 | 0 | 0° | 0^{g} |

1/24 | π/12 | 15° | 16+2/3^{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} |

## See also

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

## References

- Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 63, 1998.
- http://mathworld.wolfram.com/Gradian.html
- 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. - Frasier, E. Lewis (February 1974), "Improving an imperfect metric system",
*Bulletin of the Atomic Scientists*: 9ff. 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*, NBS Technical Note 915, U.S. Department of Commerce, National Bureau of Commerce, Institute for Applied Technology, Center for Building Technology,The term "Celsius" was adopted instead of the more familiar "centigrade" because in France the word centigrade has customarily been applied to angles.

- 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
- http://www.hpmuseum.org/forum/thread-4492.html