A circle is a
shape
A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type.
A pl ...
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, Inner Hebrides, Scotland
* Point ...
in a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
that are at a given distance from a given point, the
centre
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
. 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. The distance between any point of the circle and the centre is called the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
. Usually, the radius is required to be a positive number. A circle with
(a single point) is a
degenerate case
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
T ...
. This article is about circles in
Euclidean geometry, and, in particular, the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
, 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 (ge ...
that divides the plane into two
regions: an interior and an exterior. 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''.
A circle may also be defined as a special kind of
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
in which the two
foci
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
are coincident, the
eccentricity is 0, and the
semi-major and semi-minor axes are equal; or the two-dimensional shape enclosing the most area per unit perimeter squared, using
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
.
Euclid's definition
Topological definition
In the field of
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, a circle is not limited to the geometric concept, but to all of its
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
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, an ambient isotopy, also called an ''h-isotopy'', is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one ...
).
Terminology
*
Annulus
Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to:
Human anatomy
* ''Anulus fibrosus disci intervertebralis'', spinal structure
* Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
: a ring-shaped object, the region bounded by two
concentric
In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point ...
circles.
*
Arc
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
: 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 ( ...
: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
*
Circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to ...
: the length of one circuit along the circle, or the distance around the circle.
*
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
: 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: the region common to (the intersection of) two overlapping discs.
* Passant: a
coplanar 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
Sector may refer to:
Places
* Sector, West Virginia, U.S.
Geometry
* Circular sector, the portion of a disc enclosed by two radii and a circular arc
* Hyperbolic sector, a region enclosed by two radii and a hyperbolic arc
* Spherical sector, a po ...
: 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: 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: 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, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
: 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 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'', ''Odyssey'', and Homeric Hymns. It is a literary dialect of Ancient Greek consisting mainly of Ionic, with some Aeolic forms, a few from Arcadocypriot, and ...
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 clowns, acrobats, trained animals, trapeze acts, musicians, dancers, hoopers, tightrope walkers, jugglers, magicians, ventriloquists, and unicyclist ...
'' and ''
circuit'' 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
A wheel is a circular component that is intended to rotate on an axle Bearing (mechanical), bearing. The wheel is one of the key components of the wheel and axle which is one of the Simple machine, six simple machines. Wheels, in conjunction wi ...
, which, with related inventions such as
gear
A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic pr ...
s, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry,
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
and
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
.
Early
science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
, particularly
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
astrology and astronomy
Astrology and astronomy were archaically treated together ( la, astrologia), but gradually distinguished through the Late Middle Ages into the Age of Reason. Developments in 17th century philosophy resulted in astrology and astronomy operating ...
, 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 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, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's
Seventh Letter
The ''Seventh Letter of Plato'' is an epistle that tradition has ascribed to Plato. It is by far the longest of the epistles of Plato and gives an autobiographical
An autobiography, sometimes informally called an autobio, is a self-writte ...
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 –
Lindemann Lindemann is a German surname.
Persons
Notable people with the surname include:
Arts and entertainment
* Elisabeth Lindemann, German textile designer and weaver
* Jens Lindemann, trumpet player
* Julie Lindemann, American photographer
* Maggie ...
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 constant approximately equal to 3.141592654. Thus the circumference ''C'' is related to the radius ''r'' and diameter ''d'' by:
:
Area enclosed
As proved by
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
, 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
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
.
Equations
Cartesian coordinates
;Equation of a circle
In an ''x''–''y''
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, the circle with centre
coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
(''a'', ''b'') and radius ''r'' is the set of all points (''x'', ''y'') such that
:
This
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
, 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 between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
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
:
;Parametric form
The equation can be written in
parametric form using the
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s sine and cosine as
:
:
where ''t'' is a
parametric variable in the range 0 to 2, interpreted geometrically as the
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
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
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
of the line passing through the centre parallel to the ''x'' axis (see
Tangent half-angle substitution
In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of x into an ordinary rational function of t by setting t = \tan \tfra ...
). 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
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
, each
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
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 In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle.
Coordinates ...
.
Polar coordinates
In
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
, 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., ''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 . When , or when the origin lies on the circle, the equation becomes
:
In the general case, the equation can be solved for ''r'', giving
:
Note that without the ± sign, the equation would in some cases describe only half a circle.
Complex plane
In the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, 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
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
.
Tangent lines
The
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
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
:
or
:
If , then the slope of this line is
:
This can also be found using
implicit differentiation.
When the centre of the circle is at the origin, then the equation of the tangent line becomes
:
and its slope is
:
Properties
* The circle is the shape with the largest area for a given length of perimeter (see
Isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
).
* The circle is a highly symmetric shape: every line through the centre forms a line of
reflection symmetry, and it has
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
around the centre for every angle. Its
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
is the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
O(2,''R''). The group of rotations alone is the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
T.
* All circles are
similar.
** A circle circumference and radius are
proportional.
** The
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
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, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
.
** Thought of as a
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
of the
unit sphere, it becomes the
Riemannian circle
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle with a characteristic length. It is the circle equipped with the ''intrinsic'' Riemannian metric of a compact one-dimensional manifold of total length 2, or ...
.
* 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.
Chord
* Chords are equidistant from the centre of a circle if and only if they are equal in length.
* The
perpendicular bisector 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, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
through the centre bisecting a chord is
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to the chord.
* If a central angle and an
inscribed angle
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an in ...
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
supplementary.
** For a
cyclic quadrilateral, 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 (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
is equal to the interior opposite angle.
* An inscribed angle subtended by a diameter is a right angle (see
Thales' theorem).
* 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 (
and
). That is,
, 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
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc le ...
(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
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
(since the central angle is 180°).
Sagitta
The
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:
:
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 construction
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
s 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
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...
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
perpendicular bisector of the segment .
* Construct the
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
Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
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
In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the ...
the line segment ''PC'' will bisect the
interior angle
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
''APB'', since the segments are similar:
:
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
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
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:
:
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
:
is not a circle, but rather a line.
Thus, if ''A'', ''B'', and ''C'' are given distinct points in the plane, then the
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 triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
a unique circle, called the
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
vertices.
A
tangential polygon, such as a
tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
, is any
convex polygon within which a
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
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
In geometry, a bicentric polygon is a tangential polygon (a polygon all of whose sides are tangent to an inner incircle) which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon. All triangl ...
.
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 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
A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape.
In the ...
has an equation of the form
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
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or ...
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, distance is determined by
:
In Euclidean geometry, ''p'' = 2, giving the familiar
:
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
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
is 4 in this geometry. The formula for the unit circle in taxicab geometry is
in Cartesian coordinates and
:
in polar coordinates.
A circle of radius 1 (using this distance) is the
von Neumann neighborhood
In cellular automata, the von Neumann neighborhood (or 4-neighborhood) is classically defined on a two-dimensional square lattice and is composed of a central cell and its four adjacent cells. The neighborhood is named after John von Neumann, ...
of its center.
A circle of radius ''r'' for the
Chebyshev distance
In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension. It is na ...
(
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 L
1 and L
∞ metrics does not generalize to higher dimensions.
Locus of constant sum
Consider a finite set of
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
points the vertices of the regular polygon
are taken.
The locus of points such that the sum of the
-th power of distances
to the vertices of a given regular polygon with circumradius
is constant is a circle, if
:
, where
=1,2,…,
-1;
whose center is the centroid of the
.
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
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 geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
.
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
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
, rather than an
algebraic irrational number; that is, it is not the
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of any
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
with
rational coefficients. Despite the impossibility, this topic continues to be of interest for
pseudomath
Pseudomathematics, or mathematical crankery, is a mathematics-like activity that does not adhere to the framework of rigor of formal mathematical practice. Common areas of pseudomathematics are solutions of problems proved to be unsolvable or r ...
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
Dharma wheel, a rainbow, mandalas, rose windows and so forth.
See also
*
Affine sphere
*
Apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to the ...
*
Circle fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is ...
*
Gauss circle problem
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem is ...
*
Inversion in a circle
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
*
Line–circle intersection
*
List of circle topics
This list of circle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or concretely in physical space. It does not include metaphors like "inner circle" or "circular reasoning" in ...
*
Sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
*
Three points determine a circle
In algebraic geometry, Cramer's theorem on algebraic curves gives the necessary and sufficient number of points in the real plane falling on an algebraic curve to uniquely determine the curve in non-degenerate cases. This number is
:\frac 2,
...
*
Translation of axes
Specially named circles
*
Apollonian circles
*
Archimedean circle
In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. If the arbelos is normed such that the diameter of its outer (largest) half circle has a length of 1 and ''r ...
*
Archimedes' twin circles
In geometry, the twin circles are two special circles associated with an arbelos.
An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three semicircles that have , , and as their diame ...
*
Bankoff circle
In geometry, the Bankoff circle or Bankoff triplet circle is a certain Archimedean circle that can be constructed from an arbelos; an Archimedean circle is any circle with area equal to each of Archimedes' twin circles. The Bankoff circle was firs ...
*
Carlyle circle
In mathematics, a Carlyle circle (named for Thomas Carlyle) is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of ...
*
Chromatic circle
*
Circle of antisimilitude
In inversive geometry, the circle of antisimilitude (also known as mid-circle) of two circles, ''α'' and ''β'', is a reference circle for which ''α'' and ''β'' are inverses of each other. If ''α'' and ''β'' are non-intersecting or tangen ...
*
Ford circle
In mathematics, a Ford circle is a circle with center at (p/q,1/(2q^2)) and radius 1/(2q^2), where p/q is an irreducible fraction, i.e. p and q are coprime integers. Each Ford circle is tangent to the horizontal axis y=0, and any two Ford circles ...
*
Geodesic circle A geodesic circle is either "the locus on a surface at a constant geodesic distance from a fixed point" or a curve of constant geodesic curvature.
A geodesic disk is the region on a surface bounded by a geodesic circle.
In contrast with the ordin ...
*
Johnson circles
*
Schoch circles In geometry, the Schoch circles are twelve Archimedean circles constructed by Thomas Schoch.
History
In 1979, Thomas Schoch discovered a dozen new Archimedean circles; he sent his discoveries to ''Scientific Americans "Mathematical Games" editor Ma ...
*
Woo circles
In geometry, the Woo circles, introduced by Peter Y. Woo, are a set of infinitely many Archimedean circles.
Construction
Form an arbelos with the two inner semicircles tangent at point ''C''. Let ''m'' denote any nonnegative real number. Draw tw ...
Of a triangle
*
Apollonius circle of the excircles
*
Brocard circle
In geometry, the Brocard circle (or seven-point circle) is a circle derived from a given triangle. It passes through the circumcenter and symmedian of the triangle, and is centered at the midpoint of the line segment joining them (so that this s ...
*
Excircle
*
Incircle
*
Lemoine circle
In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the cor ...
*
Lester circle
In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle.
The result is named after June Lester, who published it in 1997, and t ...
*
Malfatti circles
In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem o ...
*
Mandart circle
*
Nine-point circle
*
Orthocentroidal circle
In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and centroid at opposite ends of its diameter. This diameter also contains the triangle's nine-point center and is a subset o ...
*
Parry circle In geometry, the Parry point is a special point associated with a plane triangle. It is the triangle center designated X(111) in Clark Kimberling's Encyclopedia of Triangle Centers. The Parry point and Parry circle are named in honor of the English ...
*
Polar circle (geometry)
In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is
: \begin
r^2 & = HA\times HD=HB\times HE=HC\times HF \\
& =-4R^2\cos A \cos B \cos C=4R^2-\frac(a^2+b^2+c^2),
\end
...
*
Spieker circle
In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center, the Spieker center, in addition to being the incenter of the medial triangle, is the cen ...
*
Van Lamoen circle
In Euclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle T. It contains the circumcenters of the six triangles that are defined inside T by its three medians.
Specifically, let A, B, C be the v ...
Of certain quadrilaterals
*
Eight-point circle
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicu ...
of an orthodiagonal quadrilateral
Of a conic section
*
Director circle
In geometry, the director circle of an ellipse or hyperbola (also called the Isoptic, orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each ...
*
Directrix circle
In geometry, focuses or foci (), singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, el ...
Of a torus
*
Villarceau circles
In geometry, Villarceau circles () are a pair of circles produced by cutting a torus obliquely through the center at a special angle.
Given an arbitrary point on a torus, four circles can be drawn through it. One is in a plane parallel to the e ...
References
Further reading
*
"Circle" in The MacTutor History of Mathematics archive
External links
*
*
*
*
*
*
{{Authority control
Elementary shapes
Conic sections
Pi