Turn (angle)

The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c) or to one revolution (symbol rev or r). Common related units of frequency are cycles per second (cps) and revolutions per minute (rpm). The angular unit of the turn is useful in connection with, among other things, electromagnetic coils (e.g., transformers), rotating objects, and the winding number of curves. Divisions of a turn include the half-turn and quarter-turn, spanning a straight angle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc. In the ISQ, an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a dimensionless quantity called rotation, defined as the ratio of a given angle and a full turn. It is represented by the symbol N. Because one turn is 2\pi radians, some have proposed representing 2\pi with the single letter 𝜏 (tau).

The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to  radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c) or to one revolution (symbol rev or r). Common related units of frequency are cycles per second (cps) and revolutions per minute (rpm). The angular unit of the turn is useful in connection with, among other things, electromagnetic coils (e.g., transformers), rotating objects, and the winding number of curves. Divisions of a turn include the half-turn and quarter-turn, spanning a straight angle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc. In the ISQ, an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a dimensionless quantity called rotation, defined as the ratio of a given angle and a full turn. It is represented by the symbol N. Because one turn is 2\pi radians, some have proposed representing 2\pi with the single letter 𝜏 (tau). One turn is equal to 2\pi = \tau ≈ radians, 360 degrees, or 400 gradians. N = \frac{\varphi}{2 \pi \text{ rad}} where 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: N = \frac \varphi \text{tr} = \{ \varphi \}_\text{tr} where { } tr is the numerical value of the angle in units of turns (see ). In the ISQ/SI, rotation is used to derive rotational frequency (the rate of change of rotation with respect to time), denoted by : 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.

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