Degree (angle) - Alternative Units

Alternative Units

In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions have simpler and more "natural" properties when their arguments are expressed in radians. These considerations outweigh the convenient divisibility of the number 360. One complete turn (360°) is equal to 2π radians, so 180° is equal to π radians, or equivalently, the degree is a mathematical constant: 1° = π⁄₁₈₀.

The turn (or revolution, full circle, full rotation, cycle) is used in technology and science. 1 turn = 360°.

With the invention of the metric system, based on powers of ten, there was an attempt to define a "decimal degree" (grad or gon), so that the number of decimal degrees in a right angle would be 100 gon, and there would be 400 gon in a circle. Although this idea was abandoned already by Napoleon, some groups have continued to use it and many scientific calculators still support it.

An angular mil, which is most used in military applications, has at least three specific variants, ranging from 1⁄₆₄₀₀ to 1⁄₆₀₀₀, each approximately equal to one milliradian. However, 1⁄₆₀₀₀ used by the Russian Army originated in Imperial Russia, where an equilateral chord was divided into tenths to give a circle of 600 units (this may be seen on a protractor, circa 1900, in the St Petersberg Museum of Artillery).