TheInfoList

A circle is a
shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. ...

consisting of all
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
in a
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
that are at a given distance from a given point, the
centre Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
. The distance between any point of the circle and the centre is called the
radius In classical geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

. Usually, the radius is required to be a positive number (Circle with $r=0$ is a degenerated case). This article is about circles in
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a sma ...
, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a
simple Simple or SIMPLE may refer to: * Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

closed
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

that divides the plane into two
regions In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and planets. The first person to use the wor ...
: an
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
and an
exterior In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric obje ...

. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a
disc Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * ''Disc'' (magazine), ...
. A circle may also be defined as a special kind of
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

in which the two
foci FOCUS is a fourth-generation programming language (4GL) computer programming programming language, language and development environment that is used to build database queries. Produced by Information Builders Inc., it was originally developed for d ...
are coincident, the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off- center, in geometry * Eccentricity (graph theory) of a ...
is 0, and the
semi-major and semi-minor axes In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
are equal; or the two-dimensional shape enclosing the most area per unit perimeter squared, using
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...
.

# Topological definition

In the field of
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, a circle isn't limited to the geometric concept, but to all of its
homeomorphism and a donut (torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of ...
s. Two topological circles are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an
ambient isotopy In the mathematical subject of topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathem ...
).

# Terminology

*
Annulus Annulus (or anulus) or annular may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus communis'', around the optic nerve * Annular ligament (disam ...
: a ring-shaped object, the region bounded by two
concentric In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

circles. * Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle. * Centre: the point equidistant from all points on the circle. *
Chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord (ast ...
: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments. *
Circumference In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
: the length of one circuit along the circle, or the distance around the circle. *
Diameter In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

: 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. *
Lens A lens is a transmissive optical Optics is the branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, s ...
: the region common to (the intersection of) two overlapping discs. * Passant: a
coplanarIn geometry, a set of points in space are coplanar if there exists a geometric Plane (mathematics), plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear points, non-collinear, t ...

straight line that has no point in common with the circle. * 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. * Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center and the endpoints of the radii. *
Segment Segment or segmentation may refer to: Biology *Segmentation (biology), the division of body plans into a series of repetitive segments **Segmentation in the human nervous system *Internodal segment, the portion of a nerve fiber between two Nodes of ...

: 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 center of the circle to which their arc belongs to. * Secant: an extended chord, a coplanar straight line, intersecting a circle in two points. *
Semicircle In (and more specifically ), a semicircle is a one-dimensional of points that forms half of a . The full of a semicircle always measures 180° (equivalently, , or a ). It has only one line of symmetry (). In non-technical usage, the term "semi ...

: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as center. 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

: 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.

# History

The word ''circle'' derives from the
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
κίρκος/κύκλος (''kirkos/kuklos''), itself a metathesis of the
Homeric Greek Homeric Greek is the form of the Greek language that was used by Homer in the ''Iliad'' and ''Odyssey'' and in the Homeric Hymns. It is a literary dialect of Ancient Greek consisting mainly of Ionic Greek, Ionic and Aeolic Greek, Aeolic, with a fe ...
κρίκος (''krikos''), meaning "hoop" or "ring". The origins of the words ''
circus A circus is a company of performers who put on diverse entertainment shows that may include s, , trained animals, acts, s, s, , s, , s, , and as well as other and stunt-oriented artists. The term ''circus'' also describes the performance w ...

'' and ''
circuitCircuit may refer to: Science and technology Electrical engineering * Electrical circuit, a complete electrical network with a closed-loop giving a return path for current ** Analog circuit, uses continuous signal levels ** Balanced circuit, p ...
'' are closely related. The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the
wheel File:Roue primitive.png, An early wheel made of a solid piece of wood A wheel is a circular component that is intended to rotate on an axle An axle or axletree is a central shaft for a rotating wheel or gear. On wheeled vehicles, the ...

, which, with related inventions such as
gear Cast iron mortise wheel with wooden cogs (powered by an external water wheel) meshing with a cast iron gear wheel, connected to a pulley A pulley is a wheel on an axle or shaft (mechanical engineering), shaft that is designed to support ...

s, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry,
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
and
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

. Early
science Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well as an invention. ...

, particularly
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

and
astrology and astronomy Astrology and astronomy were archaically treated together ( la, astrologia), and were only gradually separated in Western 17th century philosophy (the " Age of Reason") with the rejection of astrology. During the later part of the medieval perio ...
, 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. Some highlights in the history of the circle are: * 1700 BCE – The
Rhind papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum The British Museum, in the Bloomsbury Bloomsbury is a district in the West End of London. It is considered a fashionable residential area, and is the locatio ...
gives a method to find the area of a circular field. The result corresponds to (3.16049...) as an approximate value of

. * 300 BCE – Book 3 of Euclid's ''Elements'' deals with the properties of circles. * In
Plato Plato ( ; grc-gre, wikt:Πλάτων, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was an Classical Athens, Athenian philosopher during the Classical Greece, Classical period in Ancient Greece, founder of the Platonist school of thoug ...

'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. * 1880 CE –
LindemannLindemann is a German surname. Persons Notable people with the surname include: Arts and entertainment * Jens Lindemann, trumpet player * J. Shimon & J. Lindemann, Julie Lindemann, American photographer * Maggie Lindemann, American singer * Till ...
proves that is transcendental, effectively settling the millennia-old problem of squaring the circle.

# Analytic results

## Circumference

The ratio of a circle's circumference to its diameter is (pi), an
irrational Irrationality is cognition Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
approximately equal to 3.141592654. Thus the circumference ''C'' is related to the radius ''r'' and diameter ''d'' by: :$C = 2\pi r = \pi d.\,$

## Area enclosed

As proved by
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

, in his
Measurement of a Circle ''Measurement of a Circle'' or ''Dimension of the Circle'' ( Greek: , ''Kuklou metrēsis'') is a treatise that consists of three propositions by Archimedes, ca. 250 BCE. The treatise is only a fraction of what was a longer work. Propositions Prop ...
, 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: :$\mathrm = \pi r^2.\,$ Equivalently, denoting diameter by ''d'', :$\mathrm = \frac \approx 07854d^2,$ 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.

## Equations

### Cartesian coordinates

;Equation of a circle In an ''x''–''y''
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...
, the circle with centre
coordinates In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

(''a'', ''b'') and radius ''r'' is the set of all points (''x'', ''y'') such that : $\left(x - a\right)^2 + \left(y - b\right)^2 = r^2.$ This
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

, known as the ''equation of the circle'', follows from the
Pythagorean theorem In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

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 : $x^2 + y^2 = r^2.$ ;Parametric form The equation can be written in parametric form using the
trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s sine and cosine as : $x = a + r\,\cos t,$ : $y = b + r\,\sin t,$ where ''t'' is a parametric variable in the range 0 to 2, interpreted geometrically as the
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

that the ray from (''a'', ''b'') to (''x'', ''y'') makes with the positive ''x'' axis. An alternative parametrisation of the circle is : $x = a + r \frac,$ : $y = b + r \frac.$ In this parameterisation, the ratio of ''t'' to ''r'' can be interpreted geometrically as the
stereographic projection In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

of the line passing through the centre parallel to the ''x'' axis (see
Tangent half-angle substitution In integral calculus In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integr ...
). 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 $\left(x_1, y_1\right), \left(x_2, y_2\right), \left(x_3, y_3\right)$ not on a line is obtained by a conversion of the : : $\frac = \frac .$ ;Homogeneous form In
homogeneous coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, each
conic section In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
with the equation of a circle has the form : $x^2 + y^2 - 2axz - 2byz + cz^2 = 0.$ It can be proven that a conic section is a circle exactly when it contains (when extended to the
complex projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
) the points ''I''(1: ''i'': 0) and ''J''(1: −''i'': 0). These points are called the
circular points at infinityIn projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity 150px, The real line with the point at infinity; it is called the real projective line. In geometry ...
.

### Polar coordinates

In
polar coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, the equation of a circle is : $r^2 - 2 r r_0 \cos\left(\theta - \phi\right) + r_0^2 = a^2,$ where ''a'' is the radius of the circle, $\left(r, \theta\right)$ are the polar coordinates of a generic point on the circle, and $\left(r_0, \phi\right)$ are the polar coordinates of the centre of the circle (i.e., ''r''0 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 simply . When , or when the origin lies on the circle, the equation becomes : $r = 2 a\cos\left(\theta - \phi\right).$ In the general case, the equation can be solved for ''r'', giving : $r = r_0 \cos\left(\theta - \phi\right) \pm \sqrt.$ Note that without the ± sign, the equation would in some cases describe only half a circle.

### Complex plane

In the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, a circle with a centre at ''c'' and radius ''r'' has the equation : $, z - c, = r.$ In parametric form, this can be written as : $z = re^ + c.$ The slightly generalised equation : $pz\overline + gz + \overline = q$ for real ''p'', ''q'' and complex ''g'' is sometimes called a
generalised circle A generalized circle, also referred to as a "cline" or "circline", is a straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight ...
. This becomes the above equation for a circle with $p = 1,\ g = -\overline,\ q = r^2 - , c, ^2$, since $, z - c, ^2 = z\overline - \overlinez - c\overline + c\overline$. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

.

## Tangent lines

The
tangent line In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

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 (''x''1, ''y''1), so it has the form . Evaluating at (''x''1, ''y''1) determines the value of ''c'', and the result is that the equation of the tangent is : $\left(x_1 - a\right)x + \left(y_1 - b\right)y = \left(x_1 - a\right)x_1 + \left(y_1 - b\right)y_1,$ or : $\left(x_1 - a\right)\left(x - a\right) + \left(y_1 - b\right)\left(y - b\right) = r^2.$ If , then the slope of this line is : $\frac = -\frac.$ This can also be found using
implicit differentiation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. When the centre of the circle is at the origin, then the equation of the tangent line becomes : $x_1 x + y_1 y = r^2,$ and its slope is : $\frac = -\frac.$

# Properties

* The circle is the shape with the largest area for a given length of perimeter (see Isoperimetric inequality). * The circle is a highly symmetric shape: every line through the centre forms a line of
reflection symmetry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, and it has
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related ...
around the centre for every angle. Its
symmetry group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...
is the
orthogonal group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
O(2,''R''). The group of rotations alone is the
circle group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
T. * All circles are similar. ** A circle circumference and radius are
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compared ...
. ** The
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...
enclosed and the square of its radius are proportional. ** The constants of proportionality are 2 and respectively. * The circle that is centred at the origin with radius 1 is called the
unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

. ** Thought of as a
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

of the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center (geometry), center. More generally, it is the Locus (mathematics), set of points of distance 1 from a fixed central point, where different norm (mathematics), norm ...
, it becomes the
Riemannian circleImage:Sphere halve.png, A great circle divides the sphere in two equal sphere, hemispheres In metric space theory and Riemannian geometry, the Riemannian circle is a great circle equipped with its great-circle distance. It is the circle equipped wi ...
. * Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See
circumcircle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

.

## Chord

* Chords are equidistant from the centre of a circle if and only if they are equal in length. * The
perpendicular bisector In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are: ** A perpendicular line from the centre of a circle bisects the chord. ** The
line segment In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

through the centre bisecting a chord is
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

to the chord. * If a central angle and an
inscribed angle In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle. * If two angles are inscribed on the same chord and on the same side of the chord, then they are equal. * If two angles are inscribed on the same chord and on opposite sides of the chord, then they are . ** For a
cyclic quadrilateral In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's metho ...

, the
exterior angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a Simple polygon, simple (non-self-intersecting) polygon, regardless of whether it is Polygon#Convexity and non-convexity, convex or non-conve ...
is equal to the interior opposite angle. * An inscribed angle subtended by a diameter is a right angle (see
Thales' theorem In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

). * The diameter is the longest chord of the circle. ** Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB. * If the intersection of any two chords divides one chord into lengths ''a'' and ''b'' and divides the other chord into lengths ''c'' and ''d'', then . * If the intersection of any two perpendicular chords divides one chord into lengths ''a'' and ''b'' and divides the other chord into lengths ''c'' and ''d'', then equals the square of the diameter. * The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point and is given by 8''r''2 − 4''p''2, where ''r'' is the circle radius, and ''p'' is the distance from the centre point to the point of intersection. * The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.

## Tangent

* A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle. * A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle. * Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length. * If a tangent at ''A'' and a tangent at ''B'' intersect at the exterior point ''P'', then denoting the centre as ''O'', the angles ∠''BOA'' and ∠''BPA'' are supplementary. * If ''AD'' is tangent to the circle at ''A'' and if ''AQ'' is a chord of the circle, then .

## Theorems

* The chord theorem states that if two chords, ''CD'' and ''EB'', intersect at ''A'', then . * If two secants, ''AE'' and ''AD'', also cut the circle at ''B'' and ''C'' respectively, then (corollary of the chord theorem). * A tangent can be considered a limiting case of a secant whose ends are coincident. If a tangent from an external point ''A'' meets the circle at ''F'' and a secant from the external point ''A'' meets the circle at ''C'' and ''D'' respectively, then (tangent–secant theorem). * The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord (tangent chord angle). * If the angle subtended by the chord at the centre is 90 °, then , where ''ℓ'' is the length of the chord, and ''r'' is the radius of the circle. * If two secants are inscribed in the circle as shown at right, then the measurement of angle ''A'' is equal to one half the difference of the measurements of the enclosed arcs ($\overset$ and $\overset$). That is, $2\angle = \angle - \angle$, where ''O'' is the centre of the circle (secant–secant theorem).

## Inscribed angles

An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180°).

## Sagitta

The Sagitta (geometry), sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle. Given the length ''y'' of a chord and the length ''x'' of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines: : $r = \frac + \frac.$ Another proof of this result, which relies only on two chord properties given above, is as follows. Given a chord of length ''y'' and with sagitta of length ''x'', since the sagitta intersects the midpoint of the chord, we know that it is a part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is () in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (. Solving for ''r'', we find the required result.

# Compass and straightedge constructions

There are many compass-and-straightedge constructions resulting in circles. The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the Compass (drawing tool), compass on the centre point, the movable leg on the point on the circle and rotate the compass.

## Construction with given diameter

* Construct the midpoint of the diameter. * Construct the circle with centre passing through one of the endpoints of the diameter (it will also pass through the other endpoint).

## Construction through three noncollinear points

* Name the points , and , * Construct the Bisection#Line segment bisector, perpendicular bisector of the segment . * Construct the Bisection#Line segment bisector, perpendicular bisector of the segment . * Label the point of intersection of these two perpendicular bisectors . (They meet because the points are not collinear). * Construct the circle with centre passing through one of the points , or (it will also pass through the other two points).

# Circle of Apollonius

Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ''ratio'' (other than 1) of distances to two fixed foci, ''A'' and ''B''. (The set of points where the distances are equal is the perpendicular bisector of segment ''AB'', a line.) That circle is sometimes said to be drawn ''about'' two points. The proof is in two parts. First, one must prove that, given two foci ''A'' and ''B'' and a ratio of distances, any point ''P'' satisfying the ratio of distances must fall on a particular circle. Let ''C'' be another point, also satisfying the ratio and lying on segment ''AB''. By the angle bisector theorem the line segment ''PC'' will bisect the interior angle ''APB'', since the segments are similar: :$\frac = \frac.$ Analogously, a line segment ''PD'' through some point ''D'' on ''AB'' extended bisects the corresponding exterior angle ''BPQ'' where ''Q'' is on ''AP'' extended. Since the interior and exterior angles sum to 180 degrees, the angle ''CPD'' is exactly 90 degrees; that is, a right angle. The set of points ''P'' such that angle ''CPD'' is a right angle forms a circle, of which ''CD'' is a diameter. Second, see for a proof that every point on the indicated circle satisfies the given ratio.

## Cross-ratios

A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If ''A'', ''B'', and ''C'' are as above, then the circle of Apollonius for these three points is the collection of points ''P'' for which the absolute value of the cross-ratio is equal to one: : $\big, \left[A, B; C, P\right]\big, = 1.$ Stated another way, ''P'' is a point on the circle of Apollonius if and only if the cross-ratio is on the unit circle in the complex plane.

## Generalised circles

If ''C'' is the midpoint of the segment ''AB'', then the collection of points ''P'' satisfying the Apollonius condition :$\frac = \frac$ is not a circle, but rather a line. Thus, if ''A'', ''B'', and ''C'' are given distinct points in the plane, then the Locus (mathematics), locus of points ''P'' satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius.

# Inscription in or circumscription about other figures

In every triangle a unique circle, called the Incircle and excircles of a triangle, incircle, can be inscribed such that it is tangent to each of the three sides of the triangle. About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three Vertex (geometry), vertices. A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a inscribed circle, circle can be inscribed that is tangent to each side of the polygon. Every regular polygon and every triangle is a tangential polygon. A cyclic polygon is any convex polygon about which a circumcircle, circle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon. A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.

# Limiting case of other figures

The circle can be viewed as a Limiting case (mathematics), limiting case of each of various other figures: * A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two fixed points (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a different special case of a Cartesian oval in which one of the weights is zero. * A superellipse has an equation of the form $\left, \frac\^n\! + \left, \frac\^n\! = 1$ for positive ''a'', ''b'', and ''n''. A supercircle has . A circle is the special case of a supercircle in which . * A Cassini oval is a set of points such that the product of the distances from any of its points to two fixed points is a constant. When the two fixed points coincide, a circle results. * A curve of constant width is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of figure.

# In other ''p''-norms

Defining a circle as the set of points with a fixed distance from a point, different shapes can be considered circles under different definitions of distance. In p-norm, ''p''-norm, distance is determined by :$\left\, x \right\, _p = \left\left( , x_1, ^p + , x_2, ^p + \dotsb + , x_n, ^p \right\right) ^ .$ In Euclidean geometry, ''p'' = 2, giving the familiar :$\left\, x \right\, _2 = \sqrt .$ In taxicab geometry, ''p'' = 1. Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes. While each side would have length $\sqrtr$ using a Euclidean metric, where ''r'' is the circle's radius, its length in taxicab geometry is 2''r''. Thus, a circle's circumference is 8''r''. Thus, the value of a geometric analog to $\pi$ is 4 in this geometry. The formula for the unit circle in taxicab geometry is $, x, + , y, = 1$ in Cartesian coordinates and :$r = \frac$ in polar coordinates. A circle of radius 1 (using this distance) is the von Neumann neighborhood of its center. A circle of radius ''r'' for the Chebyshev distance (Lp space, L metric) on a plane is also a square with side length 2''r'' parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between L1 and L metrics does not generalize to higher dimensions.

# Locus of constant sum

Consider a finite set of $n$ points in the plane. The locus of points such that the sum of the squares of the distances to the given points is constant is a circle, whose center is at the centroid of the given points. A generalization for higher powers of distances is obtained if under $n$ points the vertices of the regular polygon $P_n$ are taken. The locus of points such that the sum of the $\left(2m\right)$-th power of distances $d_i$ to the vertices of a given regular polygon with circumradius $R$ is constant is a circle, if :$\sum_^n d_i^> nR^$, where $m$=1,2,…, $n$-1; whose center is the centroid of the $P_n$. In the case of the equilateral triangle, the loci of the constant sums of the second and fourth powers are circles, whereas for the square, the loci are circles for the constant sums of the second, fourth, and sixth powers. For the regular pentagon the constant sum of the eighth powers of the distances will be added and so forth.

# Squaring the circle

Squaring the circle is the problem, proposed by classical antiquity, ancient geometers, of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi () is a transcendental number, rather than an algebraic number, algebraic irrational number; that is, it is not the root of a function, root of any polynomial with rational number, rational coefficients. Despite the impossibility, this topic continues to be of interest for pseudomath enthusiasts.

# Significance in art and symbolism

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 Dharmachakra, Dharma wheel, a rainbow, mandalas, rose windows and so forth.

* Affine sphere * Apeirogon * Circle fitting * Inversion in a circle * List of circle topics * Sphere * Three points determine a circle * Translation of axes

## Specially named circles

* Apollonian circles * Archimedean circle * Archimedes' twin circles * Bankoff circle * Carlyle circle * Chromatic circle * Circle of antisimilitude * Ford circle * Geodesic circle * Johnson circles * Schoch circles * Woo circles

### Of a triangle

* Incircle and excircles of a triangle#Other excircle properties, Apollonius circle of the excircles * Brocard circle * Excircle * Incircle * Lemoine circle * Lester circle * Malfatti circles * Mandart circle * Nine-point circle * Orthocentroidal circle * Parry circle * Polar circle (geometry) * Spieker circle * Van Lamoen circle

* Eight-point circle of an orthodiagonal quadrilateral

### Of a conic section

* Director circle * Directrix circle

### Of a torus

* Villarceau circles