Turn (angle)

Turn
Turn
	Counterclockwise rotations about the center point where a complete rotation corresponds to an angle of rotation of 1 turn.
General information
Unit of	Plane angle
Symbol	tr, pla, rev, cyc
Conversions
1 tr in ...	... is equal to ...
radians	2π rad; ≈ 6.283185307... rad
milliradians	2000π mrad; ≈ 6283.185307... mrad
degrees	360°
gradians	400g

One turn (symbol tr or pla or N) is a unit of plane angle measurement equal to $2π$ radians, 360 degrees or 400 gradians. Thus it is the angular measure subtended by a complete circle at its center. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c)^[1] or to one revolution (symbol rev or r).^[2] Common related units of frequency are cycles per second (cps) and revolutions per minute (rpm).^[a] 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.

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. (See below for the formula.) Subdivisions of a turn include half-turns and quarter-turns, spanning a semicircle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.

Another common unit for representing angles is radians, which are usually stated in terms of $\pi$ (pi). The symbol $\pi$ , as representing one half-turn, was developed by William Jones in 1706 and then popularized by Leonhard Euler.^[3]^[4] In 2010, Michael Hartl proposed instead using the symbol $\tau$ (tau), equal to $2\pi$ and corresponding to one turn, for greater conceptual simplicity.^[5] This proposal did not initially gain widespread acceptance in the mathematical community,^[6] but the constant has become more widespread,^[7] having been added to several major programming languages and calculators.

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