# Gradian

In trigonometry, the gradian – also known as the gon (from Ancient Greek γωνία (gōnía) 'angle'), grad, or grade – 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. It is equivalent to ⁠1/400⁠ of a turn, ⁠9/10⁠ of a degree, or ⁠π/200⁠ of a radian. Measuring angles in gradians (gons) is said to employ the centesimal system of angular measurement, initiated as part of metrication and decimalisation efforts.
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. The chance of confusion was one reason for the adoption of the term Celsius to replace centigrade as the name of the temperature scale.
Gradians (gons) are principally used in surveying (especially in Europe),
and to a lesser extent in mining and geology.
The gon (gradian) is a legally recognised unit of measurement in the European Union$^{:9}$ and in Switzerland. However, this unit is not part of the International System of Units (SI).$^{:9–10}$

## 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, mining and geology. 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 (gradian), as well as radians and degrees, for their trigonometric functions. In the 2010s, some scientific calculators lack support for gradians.

### Symbol

The international standard symbol for this unit is "gon" (see ISO 31-1, Annex B). 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 | Submultiples | Submultiples | Multiples | Multiples | Multiples |
| Value | SI symbol | Name | Value | SI symbol | Name |
| 10$^{−1}$ gon | dgon | decigon | 10$^1$ gon | dagon | decagon |
| 10$^{−2}$ gon | cgon | centigon | 10$^2$ gon | hgon | hectogon |
| 10$^{−3}$ gon | mgon | milligon | 10$^3$ gon | kgon | kilogon |
| 10$^{−6}$ gon | μgon | microgon | 10$^6$ gon | Mgon | megagon |
| 10$^{−9}$ gon | ngon | nanogon | 10$^9$ gon | Ggon | gigagon |
| 10$^{−12}$ gon | pgon | picogon | 10$^{12}$ gon | Tgon | teragon |
| 10$^{−15}$ gon | fgon | femtogon | 10$^{15}$ gon | Pgon | petagon |
| 10$^{−18}$ gon | agon | attogon | 10$^{18}$ gon | Egon | exagon |
| 10$^{−21}$ gon | zgon | zeptogon | 10$^{21}$ gon | Zgon | zettagon |
| 10$^{−24}$ gon | ygon | yoctogon | 10$^{24}$ gon | Ygon | yottagon |
| 10$^{−27}$ gon | rgon | rontogon | 10$^{27}$ gon | Rgon | ronnagon |
| 10$^{−30}$ gon | qgon | quectogon | 10$^{30}$ 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° | 0$^g$ |
| ⁠1/72⁠ turn | ⁠π/36⁠ or ⁠𝜏/72⁠ rad | 5° | ⁠5+5/9⁠$^g$ |
| ⁠1/24⁠ turn | ⁠π/12⁠ or ⁠𝜏/24⁠ rad | 15° | ⁠16+2/3⁠$^g$ |
| ⁠1/16⁠ turn | ⁠π/8⁠ or ⁠𝜏/16⁠ rad | 22.5° | 25$^g$ |
| ⁠1/12⁠ turn | ⁠π/6⁠ or ⁠𝜏/12⁠ rad | 30° | ⁠33+1/3⁠$^g$ |
| ⁠1/10⁠ turn | ⁠π/5⁠ or ⁠𝜏/10⁠ rad | 36° | 40$^g$ |
| ⁠1/8⁠ turn | ⁠π/4⁠ or ⁠𝜏/8⁠ rad | 45° | 50$^g$ |
| ⁠1/2π or 𝜏⁠ turn | 1 rad | approx. 57.3° | approx. 63.7$^g$ |
| ⁠1/6⁠ turn | ⁠π/3⁠ or ⁠𝜏/6⁠ rad | 60° | ⁠66+2/3⁠$^g$ |
| ⁠1/5⁠ turn | ⁠2π or 𝜏/5⁠ rad | 72° | 80$^g$ |
| ⁠1/4⁠ turn | ⁠π/2⁠ or ⁠𝜏/4⁠ rad | 90° | 100$^g$ |
| ⁠1/3⁠ turn | ⁠2π or 𝜏/3⁠ rad | 120° | ⁠133+1/3⁠$^g$ |
| ⁠2/5⁠ turn | ⁠4π or 2𝜏 or α/5⁠ rad | 144° | 160$^g$ |
| ⁠1/2⁠ turn | π or ⁠𝜏/2⁠ rad | 180° | 200$^g$ |
| ⁠3/4⁠ turn | ⁠3π or ρ/2⁠ or ⁠3𝜏/4⁠ rad | 270° | 300$^g$ |
| 1 turn | 𝜏 or 2π rad | 360° | 400$^g$ |

## Relation to the metre

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. (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$^{:9–10}$ 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. The previous edition mentioned it only in the following footnote:
   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.