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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) [1] or to one revolution (symbol rev or r). [2]
Here, the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159. One method of deriving this formula, which originated with Archimedes , involves viewing the circle as the limit of a sequence of regular polygons with an increasing number of sides.
A page from Archimedes' Measurement of a Circle. Measurement of a Circle or Dimension of the Circle (Greek: Κύκλου μέτρησις, Kuklou metrēsis) [1] is a treatise that consists of three propositions, probably made by Archimedes, ca. 250 BCE. [2] [3] The treatise is only a fraction of what was a longer work. [4] [5]
The circle is the one-dimensional hypersphere (the 1-sphere). In topology, a circle is not limited to the geometric concept, but to all of its homeomorphisms. Two topological circles are equivalent if one can be transformed into the other via a deformation of R 3 upon itself (known as an ambient isotopy). [24]
This formula is also known as the shoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points (x 1,y 1), (x 2,y 2), and (x 3,y 3). The shoelace formula can also be used to find the areas of other polygons when their vertices are known.
By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes , who stated it in 1643. Frederick Soddy 's 1936 poem The Kiss Precise summarizes the theorem in terms of the bends (signed inverse radii) of the four circles:
The useful minimum bounding circle of three points is defined either by the circumcircle (where three points are on the minimum bounding circle) or by the two points of the longest side of the triangle (where the two points define a diameter of the circle). It is common to confuse the minimum bounding circle with the circumcircle.
One turn (corresponding to a cycle or revolution) is equal to 360°. With the invention of the metric system , based on powers of ten, there was an attempt to replace degrees by decimal "degrees" in France and nearby countries, [ note 3 ] where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° = 10 ⁄ 9 gon).