Grad (angle)

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The grad is a unit of plane angle, equivalent to ¹⁄₄₀₀ of a full circle, dividing a right angle in 100. It is also known as gon, grade, or gradian (not to be confused with the grade of an inclined surface (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 term centigrade was in use for one hundredth of a grade, and the term myriograde was in use for one ten-thousandth of a grade. This was one reason for the adoption of the term Celsius to replace Centigrade as the unit of temperature.

[edit] 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).

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 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 latter sometimes written as a superscript, similarly to a degree sign: 50^g = 45°.

[edit] Benefits

Each quadrant is assigned a range of 100 grad, which eases recognition of the four quadrants, as well as arithmetic involving perpendicular or opposite angles.

0°	=	0 grad
90°	=	100 grad
180°	=	200 grad
270°	=	300 grad
360°	=	400 grad

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 grad (clockwise from due North), say, then the direction from one's left is instantly convertible into 17 grads; while the direction from one's right is 217 grads; and the direction from behind one is 317 grads. A disadvantage is that the common angles of 30° and 60° in geometry must be expressed in fractions (33¹⁄₃ grad and 66²⁄₃ grad, respectively). Similarly, in one hour (¹⁄₂₄ day), Earth rotates by 15° or 16²⁄₃ grad.

In the decimal day of 10 decimal-hours per day, 100 decimal-minutes per hour, the Earth would have rotated 400 grads. Additionally, one grad of arc along the Earth's surface was equal to 100 kilometers of distance. This gave the Earth a rotational velocity of 40 kilometers per decimal-minute.

[edit] Conversion of some common angles

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