Turn (geometry)

Turn
Unit of	Plane angle
Symbol	tr or pla
1 tr in ...	... is equal to ...
radians	6.283185307179586... rad
radians	2π rad
degrees	360°
gons	400g

A turn is a unit of plane angle measurement equal to 2π radians, 360° or 400 gon. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot.

A turn can be subdivided in many different ways: into half turns, quarter turns, centiturns, milliturns, binary angles, points etc.

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.

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