In trigonometry, the gradian, also known as the gon (from Ancient Greek: , romanized: g?nía, 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 of a turn,^[6] of a degree, or of a radian. Measuring angles in gradians is said to employ the centesimal system of angular measurement.^{[7][8][9]:22[Note 1]}

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.^[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, 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 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,^[15][8][16]mining^[17] and geology.^[18][19] The French artillery^[who?] has used the grad for decades.^[] Today, the degree, of a turn, or the mathematically more convenient radian, 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

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 grad and grad, respectively).

Similarly, in one hour ( day), Earth rotates by 15° or 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: .

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.^[25]

Conversion

Not part of 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^[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 (?/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

References

External links

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