# Gradian

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Gon
Unit of Angle
Symbol ᵍ or gon
Unit conversions
1 ᵍ in ...... is equal to ...
turns    ${\textstyle {\frac {1}{400}}}$ turn
radians    ${\textstyle {\frac {\pi }{200}}}$ rad
degrees    ${\textstyle {\frac {9}{10}}}$°
minutes of arc    54′

The gradian is a unit of measurement of an angle, equivalent to ${\textstyle {\frac {1}{400}}}$ of a turn,[1] ${\textstyle {\frac {9}{10}}}$ of a degree, or ${\textstyle {\frac {\pi }{200}}}$ of a radian. The gradian is defined as ${\textstyle {\frac {1}{100}}}$ of the right angle (in other words, there are 100 gradians in the right angle), which implies a full turn being 400 gradians.

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.[2][3]

## 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.[4] In the 2010s some scientific calculators lack support for gradians.[5]

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°.

## Benefits

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. These observations are a consequence of the fact that the number 360 has more divisors than the number 400 does. There are eleven factors of 360 less than or equal to its square root: ${\displaystyle \{2,3,4,5,6,8,9,10,12,15,18\}}$. However, there are only seven for 400: ${\displaystyle \{2,4,5,8,10,16,20\}}$

In the 18th century, the meter was defined as the forty-millionth part of a meridian. Thus, one grad of arc along the Earth's surface corresponded to 100 kilometers of distance at the equator; 1 centigrad of arc equaled 1 kilometer; 0.1 cc (centi-centigrads) of arc equaled 1 meter.[6]

## Use in surveying

In surveying, the gradian is the default unit of measurement for angles in many parts of the world.[7] Subdivisions of the gradian used in surveying can be referred to as c (short for centigrad) and cc (effectively centi-centigrad), where 1 c = 0.01 grad and 1 cc = 0.0001 grad.

## Conversion

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

## References

1. ^ 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.
2. ^ 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.
3. ^ 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.
4. ^ 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
5. ^ 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
6. ^
7. ^ Lindeburg, Michael R. (2012), Civil Engineering Reference Manual for the PE Exam, Professional Publications, Inc., p. 78-7, ISBN 9781591263807.