A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a disc.
The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus.
Terminology
Annulus: a ring-shaped object, the region bounded by two concentric circles. Arc: any connected part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle. Centre: the point equidistant from all points on the circle. Chord: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments. Circumference: the length of one circuit along the circle, or the distance around the circle. Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius. Disc: the region of the plane bounded by a circle. In strict mathematical usage, a circle is only the boundary of the disc (or disk), while in everyday use the term "circle" may also refer to a disc. Lens: the region common to (the intersection of) two overlapping discs. Radius: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter. Usually, the radius is denoted and required to be a positive number. A circle with is a degenerate case consisting of a single point. Sector: a region bounded by two radii of equal length with a common centre and either of the two possible arcs, determined by this centre and the endpoints of the radii. Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the centre of the circle to which their arc belongs. Secant: an extended chord, a coplanar straight line, intersecting a circle in two points. Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as centre. In non-technical common usage it may mean the interior of the two-dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one. Tangent: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point"). All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries.
Etymology The word circle derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself a metathesis of the Homeric Greek κρίκος (krikos), meaning "hoop" or "ring".krikos , Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus The origins of the words circus and circuit are closely related.
History
Prehistoric people made stone circles and timber circles, and circular elements are common in petroglyphs and cave paintings. Disc-shaped prehistoric artifacts include the Nebra sky disc and jade discs called Bi.
The Egyptian Rhind papyrus, dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to (3.16049...) as an approximate value of .Chronology for 30000 BC to 500 BC . History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.
Book 3 of Euclid's Elements deals with the properties of circles. Euclid's definition of a circle is:
In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation. Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.Arthur Koestler, The Sleepwalkers: A History of Man's Changing Vision of the Universe (1959)Proclus, The Six Books of Proclus, the Platonic Successor, on the Theology of Plato Tr. Thomas Taylor (1816) Vol. 2, Ch. 2, "Of Plato"
In 1880 CE, Ferdinand von Lindemann proved that is transcendental, proving that the millennia-old problem of squaring the circle cannot be performed with straightedge and compass.Squaring the circle . History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.
With the advent of abstract art in the early 20th century, geometric objects became an artistic subject in their own right. Wassily Kandinsky in particular often used circles as an element of his compositions.
Symbolism and religious use
From the time of the earliest known civilisations – such as the Assyrians and ancient Egyptians, those in the Indus Valley and along the Yellow River in China, and the Western civilisations of ancient Greece and Rome during classical Antiquity – the circle has been used directly or indirectly in visual art to convey the artist's message and to express certain ideas. However, differences in worldview (beliefs and culture) had a great impact on artists' perceptions. While some emphasised the circle's perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of existence, but in religious traditions it represents heavenly bodies and divine spirits.
The circle signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example, a compass, a halo, the vesica piscis and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, the Dharma wheel, a rainbow, mandalas, rose windows and so forth. Magic circles are part of some traditions of Western esotericism.
Analytic results Circumference
The ratio of a circle's circumference to its diameter is (pi), an irrational constant approximately equal to 3.141592654. The ratio of a circle's circumference to its radius is . Thus the circumference C is related to the radius r and diameter d by:
Area enclosed
As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to multiplied by the radius squared:
Equivalently, denoting diameter by d,
that is, approximately 79% of the circumscribing square (whose side is of length d).
The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.
Radian
If a circle of radius is centred at the vertex of an angle, and that angle intercepts an arc of the circle with an arc length of , then the radian measure of the angle is the ratio of the arc length to the radius:
The circular arc is said to subtend the angle, known as the central angle, at the centre of the circle. One radian is the measure of the central angle subtended by a circular arc whose length is equal to its radius. The angle subtended by a complete circle at its centre is a complete angle, which measures radians, 360 degrees, or one turn.
Using radians, the formula for the arc length of a circular arc of radius and subtending a central angle of measure is
and the formula for the area of a circular sector of radius and with central angle of measure is
In the special case , these formulae yield the circumference of a complete circle and area of a complete disc, respectively.
Equations Cartesian coordinates
Equation of a circle In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that
This equation, known as the equation of the circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |x − a| and |y − b|. If the circle is centred at the origin (0, 0), then the equation simplifies to
One coordinate as a function of the other
The circle of radius with center at in the – plane can be broken into two semicircles each of which is the graph of a function, and , respectively:
for values of ranging from to .
Parametric form The equation can be written in parametric form using the trigonometric functions sine and cosine as
where t is a parametric variable in the range 0 to 2, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the positive x axis.
An alternative parametrisation of the circle is
In this parameterisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x axis (see Tangent half-angle substitution). However, this parameterisation works only if t is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted.
3-point form The equation of the circle determined by three points not on a line is obtained by a conversion of the 3-point form of a circle equation:
Homogeneous form In homogeneous coordinates, each conic section with the equation of a circle has the form
It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are called the circular points at infinity.
Polar coordinates In polar coordinates, the equation of a circle is
where a is the radius of the circle, are the polar coordinates of a generic point on the circle, and are the polar coordinates of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x axis to the line connecting the origin to the centre of the circle). For a circle centred on the origin, i.e. , this reduces to . When , or when the origin lies on the circle, the equation becomes
In the general case, the equation can be solved for r, giving
Without the ± sign, the equation would in some cases describe only half a circle.
Complex plane In the complex plane, a circle with a centre at c and radius r has the equation
In parametric form, this can be written as
The slightly generalised equation
for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with , since . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.
Tangent lines
The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form . Evaluating at (x1, y1) determines the value of c, and the result is that the equation of the tangent is
or
If , then the slope of this line is