Gradian

The gradian is a unit of plane angle, equivalent to ¹⁄₄₀₀ of a turn^[1]. It is also known as gon, grad, or grade (not to be confused with the grade of an inclined surface, i.e. gradient, slope, or pitch, the gradient in vector calculus, or the radian unit). One grad equals ⁹⁄₁₀ of a degree or ^π⁄₂₀₀ of a radian. In continental Europe, the French term centigrade was in use for one hundredth of a grad, and the term myriograde was in use for one ten-thousandth of a grad. This was one reason for the adoption of the term Celsius to replace centigrade as the unit of temperature.

1 History
2 Benefits
3 Use in surveying
4 Conversion of some common angles
5 See also
6 References
7 External links

History

The unit originated in France as the grade, along with the metric system. Due to confusion with existing grad(e) units of northern Europe, 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), and a similar terminology is still used in Danish and Icelandic.

Although attempts at a general introduction were made, the unit was really only adopted in some countries and for specialised areas, like surveying. The French artillery has used the grad for decades. The degree, ¹⁄₃₆₀ of a circle, or the mathematically more convenient radian, ¹⁄_(2π) of a circle (used in the SI system of units) are generally used instead. In the 1970s and 80s most scientific calculators offered the grad as well as radians and degrees for their trigonometric functions, but in recent years some offered degrees and radians only.

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: 50^g = 45°. "Grad" was commonly used on calculators with LCD displays, as "DEG", "GRAD", and "RAD" could all be represented as a subsection of a panel with the letters "DEGRAD".

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 gon
90°	=	100 gon
180°	=	200 gon
270°	=	300 gon
360°	=	400 gon

One advantage of this unit is that right angles are easy to add and subtract in mental arithmetic. If one is traveling on a course of 117 gon (clockwise from due North), say, then the direction from one's left is instantly convertible into 17 gon; while the direction from one's right is 217 gon; and the direction from behind one is 317 gon. A disadvantage is that the common angles of 30° and 60° in geometry must be expressed in fractions (33¹⁄₃ gon and 66²⁄₃ gon, respectively). Similarly, in one hour (¹⁄₂₄ day), Earth rotates by 15° or 16²⁄₃ gon.

Originally, 1 gon of arc along the Earth's surface was equal to 100 kilometers of distance at the equator; therefore 1 centigrad of arc equals 1 kilometer.^[2]

Gradians are also convenient when working with vectors on the complex plane. The exponent of the imaginary unit on any given vector is equal to its angle in hectogradians (100 gon) from the positive x-axis.

Use in surveying

In surveying gradian is the default unit of angles in many parts of the world. Subdivisions of gradian used in surveying are c's (1c=0.01 grad) and cc's (1 cc=0.0001 grad).

Conversion of some common angles

Units	Values
Turns	0	$\tfrac{\tau}{12}$	$\tfrac{\tau}{8}$	$\tfrac{\tau}{6}$	$\tfrac{\tau}{4}$	$\tfrac{\tau}{2}$	$\tfrac{3}{4}\tau$	$1\tau$
Degrees	0°	30°	45°	60°	90°	180°	270°	360°
Radians	0	$\tfrac{\pi}{6}$	$\tfrac{\pi}{4}$	$\tfrac{\pi}{3}$	$\tfrac{\pi}{2}$	$\pi$	$\tfrac{3\pi}{2}$	2 $\pi$
Grads	0^g	$\tfrac{100^g}{3}$	50^g	$\tfrac{200^g}{3}$	100^g	200^g	300^g	400^g