|Symbol||g, gon or grad|
|radians||?/200 rad |
? 0.0157... rad
|milliradians||5? mrad |
? 15.71... mrad
|minutes of arc||54?|
In trigonometry, the gradian, also known as the gon (from Ancient Greek: , romanized: g?nía, lit. 'angle'), grad, or grade, 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. It is equivalent to 1/400 of a turn, 9/10 of a degree, or ?/200 of a radian. Measuring angles in gradians is said to employ the centesimal system of angular measurement.[Note 1]
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.
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, mining and geology. The French artillery[who?] has used the grad for decades. Today, the degree, 1/360 of a turn, or the mathematically more convenient radian, 1/2? 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. In the 2010s, some scientific calculators lack support for gradians.
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°.
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.
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: . However, there are only seven for 400: .
|Turns||Radians||Degrees||Gradians, or gons|
|0 turn||0 rad||0°||0g|
|1/24 turn||?/12 rad||15°||16+2/3g|
|1/16 turn||?/8 rad||22.5°||25g|
|1/12 turn||?/6 rad||30°||33+1/3g|
|1/10 turn||?/5 rad||36°||40g|
|1/8 turn||?/4 rad||45°||50g|
|1/2? turn||1 rad||c. 57.3°||c. 63.7g|
|1/6 turn||?/3 rad||60°||66+2/3g|
|1/5 turn||2?/5 rad||72°||80g|
|1/4 turn||?/2 rad||90°||100g|
|1/3 turn||2?/3 rad||120°||133+1/3g|
|2/5 turn||4?/5 rad||144°||160g|
|1/2 turn||? rad||180°||200g|
|3/4 turn||3?/2 rad||270°||300g|
|1 turn||2? rad||360°||400g|
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.
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 a footnote, which said the following:
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 (?/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 , 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.
The angle through which the torsion-head must be deflected was measured in centesimal divisions of the circle
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]
It hung on a ribbon torsion element with a knob on top, graduated in 100 parts.
The term "Celsius" was adopted instead of the more familiar "centigrade" because in France the word centigrade has customarily been applied to angles.
On the approximation of the laws of the Member States relating to units of measurement and on the repeal of Directive 71/354/EEC.
On most scientific calculators, this [the unit for angles] is set by the DRG key
Scientific calculators commonly have two modes for working with angles - degrees and radians