Turn (angle)

Divisions

Many angle units are defined as a division of the turn. For example, the degree is defined such that one turn is 360 degrees.

Using metric prefixes, the turn can be divided in 100 centiturns or 1000 milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″. A protractor divided in centiturns is normally called a "percentage protractor". While percentage protractors have existed since 1922, the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962. Some measurement devices for artillery and satellite watching carry milliturn scales.

Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, which implicitly have an angular separation of ⁠1/32⁠ turn. The binary degree, also known as the binary radian (or brad), is ⁠1/256⁠ turn. The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividing one whole turn into 2ⁿ equal parts for other values of n.

Unit conversion

One turn is equal to ${\displaystyle 2\pi }$ = ${\displaystyle \tau }$ ≈ 6.283185307179586 radians, 360 degrees, or 400 gradians.

In the ISQ/SI

In the International System of Quantities (ISQ), rotation (symbol N) is a physical quantity defined as number of revolutions:

N is the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis. Its value is given by:

${\displaystyle N=\frac{\phi }{2\pi \text{ rad}}}$

where 𝜑 denotes the measure of rotational displacement.

The above definition is part of the ISQ, formalized in the international standard ISO 80000-3 (Space and time), and adopted in the International System of Units (SI).

Rotation count or number of revolutions is a quantity of dimension one, resulting from a ratio of angular displacement. It can be negative and also greater than 1 in modulus. The relationship between quantity rotation, N, and unit turns, tr, can be expressed as:

${\displaystyle N=\frac{\phi }{\text{tr}}=\{\phi {\}}_{\text{tr}}}$

where {𝜑}_tr is the numerical value of the angle 𝜑 in units of turns (see Physical quantity § Components).

In the ISQ/SI, rotation is used to derive rotational frequency (the rate of change of rotation with respect to time), denoted by n:

${\displaystyle n=\frac{\mathrm{d}N}{\mathrm{d}t}}$

The SI unit of rotational frequency is the reciprocal second (s⁻¹). Common related units of frequency are hertz (Hz), cycles per second (cps), and revolutions per minute (rpm).

The superseded version ISO 80000-3:2006 defined "revolution" as a special name for the dimensionless unit "one", which also received other special names, such as the radian. Despite their dimensional homogeneity, these two specially named dimensionless units are applicable for non-comparable kinds of quantity: rotation and angle, respectively. "Cycle" is also mentioned in ISO 80000-3, in the definition of period.