A 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.
Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points. 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.
The notion of turn is commonly used for planar rotations. Two special rotations have acquired appellations of their own: a rotation through 180° is commonly referred to as a half-turn (π radians), a rotation through 90° is referred to as a quarter-turn.
The word turn originates via Latin and French from the Greek word τόρνος (tornos – a lathe).
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. However, earlier in 1647, William Oughtred had used δ/π (delta over pi) for the ratio of the diameter to perimeter. 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.Euler adopted the symbol with that meaning in 1737, leading to its widespread use.
Percentage protractors have existed since 1922, but the terms centiturns and milliturns were introduced much later by Sir Fred Hoyle.
The German standard DIN 1315 (1974-03) proposed the unit symbol pla (from Latin: plenus angulus "full angle") for turns. Since 2011, the HP 39gII and HP Prime support the unit symbol tr for turns. In 2016, support for turns was also added to newRPL for the HP 50g.
An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to a full turn, or approximately 6.28 radians, which are expressed here using the Greek letter tau (τ).
In 1958, Albert Eagle proposed τ as a symbol for 1/2π, selecting the new symbol because π resembles two τ symbols conjoined (ττ).
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. His proposal used a "pi with three legs" symbol to denote the constant ( = 2π).
In 2010, Michael Hartl proposed to use the Greek letter tau (τ) instead of pi (π) for two reasons. First, τ is the number of radians in one turn, which allows fractions of a turn to be expressed more directly: for instance, a 3/4 turn would be represented as 3/4τ rad instead of 3/2π rad. Second, τ visually resembles π, whose association with the circle constant is unavoidable. Hartl's Tau Manifesto gives many examples of formulas that are simpler if tau is used instead of pi. However, a rebuttal was given in The Pi Manifesto, stating a variety of reasons tau should not supplant pi. A more recent version of the Tau Manifesto rebutted the rebuttal, and extended the analysis of the surface area and volume of a n-dimensional sphere to show "the geometric significance of π is the result of a mathematical pun". Namely: