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 2π 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. (See below for the formula.)

Because one turn is

2

π

{\displaystyle 2\pi }

radians, some have proposed representing

2

π

{\displaystyle 2\pi }

with the single letter 𝜏 (tau).

Unit symbols

There are several unit symbols for the turn.

EU and Switzerland

The German standard DIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin: plenus angulus 'full angle') for turns.