[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.

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

[6][7] Covered in DIN 1301-1 [de] (October 2010), the so-called Vollwinkel ('full angle') is not an SI unit.

While percentage protractors have existed since 1922,[18] the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962.

[21] 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.[22]

In 1697, David Gregory used ⁠π/ρ⁠ (pi over rho) to denote the perimeter of a circle (i.e., the circumference) divided by its radius.

The first use of the symbol π on its own with its present meaning (of perimeter divided by diameter) was in 1706 by the Welsh mathematician William Jones.

[34][35] Several people have independently proposed using 𝜏 = 2π, including:[36] In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of π, which amounts to the number of radians in half a turn, in order to make mathematics simpler and more intuitive.

[38] The same year, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2π.

[41] In 2010, Michael Hartl proposed to use the Greek letter tau to represent the circle constant: τ = 2π.

Hartl's Tau Manifesto[b] gives many examples of formulas that are asserted to be clearer where τ is used instead of π.

He also claims that the formula for circular area in terms of τ, A = ⁠1/2⁠𝜏r2, contains a natural factor of ⁠1/2⁠ arising from integration.

[67] 𝜏 has been covered in videos by Vi Hart,[68][69][70] Numberphile,[71][72][73] SciShow,[74] Steve Mould,[75][76][77] Khan Academy,[78] and 3Blue1Brown,[79][80] and it has appeared in the comics xkcd,[81][82] Saturday Morning Breakfast Cereal,[83][84][85] and Sally Forth.

[86] The Massachusetts Institute of Technology usually announces admissions on March 14 at 6:28 p.m., which is on Pi Day at Tau Time.

In the International System of Quantities (ISQ), rotation (symbol N) is a physical quantity defined as number of revolutions:[89] N is the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis.

[90][91] Rotation count or number of revolutions is a quantity of dimension one, resulting from a ratio of angular displacement.

The relationship between quantity rotation, N, and unit turns, tr, can be expressed as: where {𝜑}tr is the numerical value of the angle 𝜑 in units of turns (see Physical quantity § Components).

Common related units of frequency are hertz (Hz), cycles per second (cps), and revolutions per minute (rpm).

[d] Despite their dimensional homogeneity, these two specially named dimensionless units are applicable for non-comparable kinds of quantity: rotation and angle, respectively.

[e] The following table documents various programming languages that have implemented the circle constant for converting between turns and radians.