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A '''circle''' is a [[Shape|shape]] consisting of all [[Point (geometry)|points]] in a [[Plane (mathematics)|plane]] that are at a given distance from a given point, the<!-- This article is written using British English – see WP:ENGVAR, Wikipedia articles should remain with the original variant 'center' is the American English spelling --> [[Centre (geometry)|centre]].<!--see comment before about 'center'--> The distance between any point of the circle and the centre is called the [[Radius|radius]]. The length of a line segment connecting two points on the circle and passing through the centre is called the [[Diameter|diameter]]. A circle bounds a region of the plane called a [[Disk (mathematics)|disc]].

The circle has been known since before the beginning of recorded history. Natural circles are common, such as the [[Astronomy:Full moon|full moon]] or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as [[Physics:Gear|gear]]s, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of [[Geometry|geometry]], [[Astronomy:Astronomy|astronomy]] and [[Calculus|calculus]].

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==Terminology==
* [[Annulus (mathematics)|Annulus]]: a ring-shaped object, the region bounded by two concentric circles.
* [[Circular arc|Arc]]: any [[Connected space|connected]] part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle.
* [[Centre (geometry)|Centre]]: the point equidistant from all points on the circle.
* [[Chord (geometry)|Chord]]: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
* [[Circumference]]: the [[Length|length]] of one circuit along the circle, or the distance around the circle.
* [[Diameter]]: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
* [[Disk (mathematics)|Disc]]: the region of the plane bounded by a circle. In strict mathematical usage, a circle is only the boundary of the disc (or disk), while in everyday use the term "circle" may also refer to a disc.
* [[Lens (geometry)|Lens]]: the region common to (the intersection of) two overlapping discs.
* [[Radius]]: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter. Usually, the radius is denoted <math>r</math> and required to be a positive number. A circle with <math>r=0</math> is a [[Degeneracy (mathematics)|degenerate case]] consisting of a single point.
* [[Circular sector|Sector]]: a region bounded by two radii of equal length with a common centre and either of the two possible arcs, determined by this centre and the endpoints of the radii.
* [[Circular segment|Segment]]: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term ''segment'' is used only for regions not containing the centre of the circle to which their arc belongs.
* [[Secant line|Secant]]: an extended chord, a coplanar straight line, intersecting a circle in two points.
* [[Semicircle]]: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as centre. In non-technical common usage it may mean the interior of the two-dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.
* [[Tangent]]: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point").
All of the specified regions may be considered as ''open'', that is, not containing their boundaries, or as ''closed'', including their respective boundaries.
{{Clear}}
{| class="skin-invert-image" style="float:left;" cellspacing="0" cellpadding="0"
|-
|[[File:CIRCLE LINES.svg|right|thumb|Chord, secant, tangent, radius, and diameter]]
|[[File:Circle slices (mul).svg|right|thumb|Arc, sector, and segment]]
|}
{{Clear}}

==Etymology==
The word ''circle'' derives from the [[Greek language|Greek]] κίρκος/κύκλος (''kirkos/kuklos''), itself a metathesis of the Homeric Greek κρίκος (''krikos''), meaning "hoop" or "ring".<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dkri%2Fkos krikos] {{webarchive|url=https://web.archive.org/web/20131106164504/http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dkri%2Fkos |date=2013-11-06 }}, Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref> The origins of the words ''circus'' and ''circuit'' are closely related.

==History==
[[File:2009_07_09_camino_cielo_paradise_137.jpg|thumb|Circular cave paintings in [[Santa Barbara County, California]]]]
[[Image:Shatir500.jpg|right|thumb|200px|Circles in an old [[Arabic]] astronomical drawing.]]
Prehistoric people made [[Unsolved:Stone circle|stone circle]]s and timber circles, and circular elements are common in petroglyphs and cave paintings.<ref>{{Cite journal |last1=Simek |first1=Jan F. |last2=Cressler |first2=Alan |last3=Herrmann |first3=Nicholas P. |last4=Sherwood |first4=Sarah C. |date=2013-06-01 |title=Sacred landscapes of the south-eastern USA: prehistoric rock and cave art in Tennessee |url=https://www.cambridge.org/core/product/identifier/S0003598X00049048/type/journal_article |journal=Antiquity |language=en |volume=87 |issue=336 |pages=430–446 |doi=10.1017/S0003598X00049048 |s2cid=130296519 |issn=0003-598X|url-access=subscription }}</ref> Disc-shaped prehistoric artifacts include the [[Astronomy:Nebra sky disc|Nebra sky disc]] and jade discs called Bi.

The Egyptian Rhind papyrus, dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to {{sfrac|256|81}} (3.16049...) as an approximate value of [[Pi|{{pi}}]].<ref>[http://www-history.mcs.st-andrews.ac.uk/history/Chronology/30000BC_500BC.html#1700BC Chronology for 30000 BC to 500 BC] {{webarchive|url=https://web.archive.org/web/20080322085509/http://www-history.mcs.st-andrews.ac.uk/history/Chronology/30000BC_500BC.html |date=2008-03-22 }}. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.</ref>

Book 3 of [[Euclid's Elements|Euclid's ''Elements'']] deals with the properties of circles. Euclid's definition of a circle is:
{{quotation|A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.|author=[[Biography:Euclid|Euclid]]|title=[[Euclid's Elements#Contents|Book I]] |source=[[Euclid's Elements|Elements]]<ref>{{OL|7227282M}}</ref>{{rp|4}}}}

In [[Biography:Plato|Plato]]'s Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation. Early [[Science|science]], particularly [[Geometry|geometry]] and [[Astronomy:Astrology and astronomy|astrology and astronomy]], was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.<ref>[[Biography:Arthur Koestler|Arthur Koestler]], ''The Sleepwalkers: A History of Man's Changing Vision of the Universe'' (1959)</ref><ref>[[Biography:Proclus|Proclus]], [https://books.google.com/books?id=E1HYAAAAMAAJ ''The Six Books of Proclus, the Platonic Successor, on the Theology of Plato''] {{webarchive|url=https://web.archive.org/web/20170123072440/https://books.google.com/books?id=E1HYAAAAMAAJ |date=2017-01-23 }} Tr. Thomas Taylor (1816) Vol. 2, Ch. 2, "Of Plato"</ref>

In 1880 CE, [[Biography:Ferdinand von Lindemann|Ferdinand von Lindemann]] proved that {{pi}} is [[Transcendental number|transcendental]], proving that the millennia-old problem of [[Squaring the circle|squaring the circle]] cannot be performed with straightedge and compass.<ref>[http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Squaring_the_circle.html Squaring the circle] {{webarchive|url=https://web.archive.org/web/20080624144640/http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Squaring_the_circle.html |date=2008-06-24 }}. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.</ref>

With the advent of abstract art in the early 20th century, geometric objects became an artistic subject in their own right. Wassily Kandinsky in particular often used circles as an element of his compositions.<ref>{{Cite web |title=Circles in a Circle |url=https://philamuseum.org/collection/object/51019 |access-date=2023-12-28 |website=Philadelphia Museum of Art |language=en}}</ref><ref>{{Cite web |last=Lesso|first=Rosie|date=2022-06-15 |title=Why Did Wassily Kandinsky Paint Circles? |url=https://www.thecollector.com/why-did-wassily-kandinsky-paint-circles/ |access-date=2023-12-28 |website=TheCollector |language=en}}</ref>

===Symbolism and religious use===
[[Image:God the Geometer.jpg|thumb|right|200px|
The compass in this 13th-century manuscript is a symbol of God's act of [[Astronomy:Creation myth|Creation]]. Notice also the circular shape of the halo.]]
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 [[Unsolved:Dharmachakra|Dharma wheel]], a rainbow, mandalas, rose windows and so forth.<ref>{{cite encyclopedia | editor-first=Jean-François | editor-last = Charnier | title = The Circle from East to West | first = Yahya | last = Abdullahi | encyclopedia = The Louvre Abu Dhabi: A World Vision of Art | date = October 29, 2019 | isbn=9782370741004 | publisher=Rizzoli International Publications, Incorporated}}</ref> Magic circles are part of some traditions of [[Philosophy:Western esotericism|Western esotericism]].

==Analytic results==
===Circumference===
{{main|Circumference}}
The ratio of a circle's circumference to its diameter is {{pi}} (pi), an [[Irrational number|irrational]] [[Mathematical constant|constant]] approximately equal to 3.141592654. The ratio of a circle's circumference to its radius is {{math|2{{pi}}}}.{{efn|Also known as {{tau}} (tau).}} Thus the circumference ''C'' is related to the radius ''r'' and diameter ''d'' by:
<math display="block">C = 2\pi r = \pi d.</math>

===Area enclosed===
[[Image:Circle Area.svg|thumb|Area enclosed by a circle = {{pi}} × area of the shaded square]]
{{Main article|Area of a circle}}
As proved by [[Biography:Archimedes|Archimedes]], in his [[Measurement of a Circle]], the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,<ref>{{citation|first=Victor J.|last=Katz|title=A History of Mathematics / An Introduction|edition=2nd|year=1998|publisher=Addison Wesley Longman|isbn=978-0-321-01618-8|page=[https://archive.org/details/historyofmathema00katz/page/108 108]|url-access=registration|url=https://archive.org/details/historyofmathema00katz/page/108}}</ref> which comes to {{pi}} multiplied by the radius squared:
<math display="block">\mathrm{Area} = \pi r^2.</math>

Equivalently, denoting diameter by ''d'',
<math display="block">\mathrm{Area} = \frac{\pi d^2}{4} \approx 0.7854 d^2,</math>
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|isoperimetric inequality]].

===Radian===
{{main|Radian}}
If a circle of radius {{mvar|r}} is centred at the [[Vertex (geometry)|vertex]] of an [[Angle|angle]], and that angle intercepts an [[Circular arc|arc of the circle]] with an [[Arc length|arc length]] of {{mvar|s}}, then the [[Radian|radian]] measure {{theta}} of the angle is the ratio of the arc length to the radius:
<math display="block">\theta = \frac{s}{r}.</math>

The circular arc is said to subtend the angle, known as the [[Central angle|central angle]], at the centre of the circle. One radian is the measure of the central angle subtended by a circular arc whose length is equal to its radius. The angle subtended by a complete circle at its centre is a complete angle, which measures {{math|2{{pi}}}} radians, 360 [[Degree (angle)|degrees]], or one [[Turn (angle)|turn]].

Using radians, the formula for the arc length {{mvar|s}} of a circular arc of radius {{mvar|r}} and subtending a central angle of measure {{theta}} is
<math display="block">s = \theta r,</math>

and the formula for the area {{mvar|A}} of a [[Circular sector|circular sector]] of radius {{mvar|r}} and with central angle of measure {{theta}} is
<math display="block">A = \frac{1}{2} \theta r^2.</math>

In the special case {{math|1={{theta}} = 2{{pi}}}}, these formulae yield the circumference of a complete circle and area of a complete disc, respectively.

===Equations===
==== Cartesian coordinates ====
[[Image:Circle center a b radius r.svg|thumb|right|Circle of radius ''r''&nbsp;=&nbsp;1, centre (''a'',&nbsp;''b'') =&nbsp;(1.2,&nbsp;−0.5)]]
===== Equation of a circle =====
In an ''x''–''y'' [[Cartesian coordinate system]], the circle with centre [[Coordinate system|coordinates]] (''a'', ''b'') and radius ''r'' is the set of all points (''x'', ''y'') such that
<math display="block">(x - a)^2 + (y - b)^2 = r^2.</math>

This [[Equation|equation]], known as the ''equation of the circle'', follows from the [[Pythagorean theorem]] applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |''x'' − ''a''| and |''y'' − ''b''|. If the circle is centred at the origin (0,&nbsp;0), then the equation simplifies to
<math display="block">x^2 + y^2 = r^2.</math>

===== One coordinate as a function of the other =====
[[File:Circle derivative.png|thumb| Upper semicircle with radius {{math|1}} and center {{math|(0, 0)}} and its derivative.]]

The circle of radius {{tmath|r}} with center at {{tmath|(x_0, y_0)}} in the {{tmath|x}}–{{tmath|y}} plane can be broken into two semicircles each of which is the [[Graph of a function|graph of a function]], {{tmath|y_+(x)}} and {{tmath|y_-(x)}}, respectively:
<math display=block>\begin{align}
y_+(x) = y_0 + \sqrt{ r^2 - (x - x_0)^2}, \\[5mu]
y_-(x) = y_0 - \sqrt{ r^2 - (x - x_0)^2},
\end{align}</math>
for values of {{tmath|x}} ranging from {{tmath|x_0 - r}} to {{tmath|x_0 + r}}.

===== Parametric form =====
The equation can be written in [[Parametric equation|parametric form]] using the [[Trigonometric function|trigonometric function]]s sine and cosine as
<math display="block">\begin{align}
x &= a + r\,\cos t, \\
y &= b + r\,\sin t,
\end{align}</math>
where ''t'' is a parametric variable in the range 0 to 2{{pi}}, interpreted geometrically as the [[Angle|angle]] that the ray from (''a'',&nbsp;''b'') to (''x'',&nbsp;''y'') makes with the positive ''x''&nbsp;axis.

An alternative parametrisation of the circle is
<math display="block">\begin{align}
x &= a + r \frac{1 - t^2}{1 + t^2}, \\
y &= b + r \frac{2t}{1 + t^2}.
\end{align}</math>

In this parameterisation, the ratio of ''t'' to ''r'' can be interpreted geometrically as the [[Stereographic projection|stereographic projection]] of the line passing through the centre parallel to the ''x''&nbsp;axis (see [[Tangent half-angle substitution]]). However, this parameterisation works only if ''t'' is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted.

===== 3-point form =====
The equation of the circle determined by three points <math>(x_1, y_1), (x_2, y_2), (x_3, y_3)</math> not on a line is obtained by a conversion of the [[Ellipse#Circles|''3-point form of a circle equation'']]:
<math display="block">
\frac{({\color{green}x} - x_1)({\color{green}x} - x_2) + ({\color{red}y} - y_1)({\color{red}y} - y_2)}
     {({\color{red}y} - y_1)({\color{green}x} - x_2) - ({\color{red}y} - y_2)({\color{green}x} - x_1)} =
\frac{(x_3 - x_1)(x_3 - x_2) + (y_3 - y_1)(y_3 - y_2)}
     {(y_3 - y_1)(x_3 - x_2) - (y_3 - y_2)(x_3 - x_1)}.</math>

===== Homogeneous form =====
In [[Homogeneous coordinates|homogeneous coordinates]], each [[Conic section|conic section]] with the equation of a circle has the form
<math display="block">x^2 + y^2 - 2axz - 2byz + cz^2 = 0.</math>

It can be proven that a conic section is a circle exactly when it contains (when extended to the [[Complex projective plane|complex projective plane]]) the points ''I''(1: ''i'': 0) and ''J''(1:&nbsp;−''i'':&nbsp;0). These points are called the [[Circular points at infinity|circular points at infinity]].

====Polar coordinates====
In [[Polar coordinates|polar coordinates]], the equation of a circle is
<math display="block">r^2 - 2 r r_0 \cos(\theta - \phi) + r_0^2 = a^2,</math>

where ''a'' is the radius of the circle, <math>(r, \theta)</math> are the polar coordinates of a generic point on the circle, and <math>(r_0, \phi)</math> are the polar coordinates of the centre of the circle (i.e., ''r''<sub>0</sub> is the distance from the origin to the centre of the circle, and ''φ'' is the anticlockwise angle from the positive ''x''&nbsp;axis to the line connecting the origin to the centre of the circle). For a circle centred on the origin, i.e. {{nowrap|''r''<sub>0</sub> {{=}} 0}}, this reduces to {{nowrap|''r'' {{=}} ''a''}}. When {{nowrap|''r''<sub>0</sub> {{=}} ''a''}}, or when the origin lies on the circle, the equation becomes
<math display="block">r = 2 a\cos(\theta - \phi).</math>

In the general case, the equation can be solved for ''r'', giving
<math display="block">r = r_0 \cos(\theta - \phi) \pm \sqrt{a^2 - r_0^2 \sin^2(\theta - \phi)}.</math>
Without the ± sign, the equation would in some cases describe only half a circle.

====Complex plane====
In the [[Complex plane|complex plane]], a circle with a centre at ''c'' and radius ''r'' has the equation
<math display="block">|z - c| = r.</math>

In parametric form, this can be written as
<math display="block">z = re^{it} + c.</math>

The slightly generalised equation
<math display="block">pz\overline{z} + gz + \overline{gz} = q</math>

for real ''p'', ''q'' and complex ''g'' is sometimes called a [[Generalised circle|generalised circle]]. This becomes the above equation for a circle with <math>p = 1,\ g = -\overline{c},\ q = r^2 - |c|^2</math>, since <math>|z - c|^2 = z\overline{z} - \overline{c}z - c\overline{z} + c\overline{c}</math>. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a [[Line (geometry)|line]].

===Tangent lines===
{{Main article|Tangent lines to circles}}
The [[Tangent line|tangent line]] through a point ''P'' on the circle is perpendicular to the diameter passing through ''P''. If {{nowrap|P {{=}} (''x''<sub>1</sub>, ''y''<sub>1</sub>)}} and the circle has centre (''a'', ''b'') and radius ''r'', then the tangent line is perpendicular to the line from (''a'', ''b'') to (''x''<sub>1</sub>, ''y''<sub>1</sub>), so it has the form {{nowrap|(''x''<sub>1</sub> − ''a'')''x'' + (''y''<sub>1</sub> − ''b'')''y'' {{=}} ''c''}}. Evaluating at (''x''<sub>1</sub>, ''y''<sub>1</sub>) determines the value of ''c'', and the result is that the equation of the tangent is
<math display="block">(x_1 - a)x + (y_1 - b)y = (x_1 - a)x_1 + (y_1 - b)y_1,</math>
or
<math display="block">(x_1 - a)(x - a) + (y_1 - b)(y - b) = r^2.</math>

If {{nowrap|''y''<sub>1</sub> ≠ ''b''}}, then the slope of this line is
<math display="block">\frac{dy}{dx} = -\frac{x_1 - a}{y_1 - b}.</math>

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
<math display="block">x_1 x + y_1 y = r^2,</math>
and its slope is
<math display="block">\frac{dy}{dx} = -\frac{x_1}{y_1}.</math>

==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|reflection symmetry]], and it has [[Rotational symmetry|rotational symmetry]] around the centre for every angle. Its [[Symmetry group|symmetry group]] is the [[Orthogonal group|orthogonal group]] O(2,''R''). The group of rotations alone is the [[Circle group|circle group]] '''T'''.
* All circles are [[Similarity (geometry)|similar]].<ref>{{cite journal
 | last = Richeson | first = David
 | arxiv = 1303.0904
 | doi = 10.4169/college.math.j.46.3.162
 | issue = 3
 | journal = The College Mathematics Journal
 | mr = 3413900
 | pages = 162–171
 | title = Circular reasoning: who first proved that {{mvar|C}} divided by {{var|d}} is a constant?
 | volume = 46
 | year = 2015}}</ref>
** A circle circumference and radius are [[Proportionality (mathematics)|proportional]].
** The area enclosed and the square of its radius are proportional.
** The constants of proportionality are 2{{pi}} and {{pi}} respectively.
* The circle that is centred at the origin with radius 1 is called the [[Unit circle|unit circle]].
** Thought of as a [[Great circle|great circle]] of the [[Unit sphere|unit sphere]], it becomes the [[Riemannian circle]].
* 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|line segment]] through the centre bisecting a chord is [[Perpendicular|perpendicular]] to the chord.
* If a central angle and an [[Inscribed angle|inscribed angle]] 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|cyclic quadrilateral]], the exterior angle 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 [[Intersecting chords theorem|intersection of any two chords]] divides one chord into lengths ''a'' and ''b'' and divides the other chord into lengths ''c'' and ''d'', then {{nowrap|''ab'' {{=}} ''cd''}}.
* 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 {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup>}} equals the square of the diameter.<ref>Posamentier and Salkind, ''Challenging Problems in Geometry'', Dover, 2nd edition, 1996: pp. 104–105, #4–23.</ref>
* 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''<sup>2</sup> − 4''p''<sup>2</sup>, where ''r'' is the circle radius, and ''p'' is the distance from the centre point to the point of intersection.<ref>''College Mathematics Journal'' 29(4), September 1998, p. 331, problem 635.</ref>
* 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.<ref>Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007.</ref>{{rp|p.71}}

===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 {{nowrap|∠''DAQ'' {{=}} {{sfrac|1|2}}arc(''AQ'')}}.

===Theorems===
[[Image:Secant-Secant Theorem.svg|thumb|right|Secant–secant theorem]]
{{See also|Power of a point}}
* The chord theorem states that if two chords, ''CD'' and ''EB'', intersect at ''A'', then {{nowrap|''AC'' × ''AD'' {{=}} ''AB'' × ''AE''}}.
* If two secants, ''AE'' and ''AD'', also cut the circle at ''B'' and ''C'' respectively, then {{nowrap|''AC'' × ''AD'' {{=}} ''AB'' × ''AE''}} (corollary of the chord theorem).
* {{anchor|Tangent-secant 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 {{nowrap|''AF''<sup>2</sup> {{=}} ''AC'' × ''AD''}} (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[[Degree (angle)|°]], then {{nowrap|''ℓ'' {{=}} ''r'' √2}}, where ''ℓ'' is the length of the chord, and ''r'' is the radius of the circle.
* {{anchor|Secant-secant theorem}}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 (<math>\overset{\frown}{DE}</math> and <math>\overset{\frown}{BC}</math>). That is, <math>2\angle{CAB} = \angle{DOE} - \angle{BOC}</math>, where ''O'' is the centre of the circle (secant–secant theorem).

===Inscribed angles===
{{See also|Inscribed angle theorem}}
[[Image:inscribed angle theorem.svg|thumb|200px|right|Inscribed-angle theorem]]
An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding [[Central angle|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|right angle]] (since the central angle is 180°).

{{clear}}

===Sagitta===
[[Image:circle Sagitta.svg|thumb|277px|right|The sagitta is the vertical segment.]]
The [[Sagitta (geometry)|sagitta]] (also known as the [[Versine|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:
<math display="block">r = \frac{y^2}{8x} + \frac{x}{2}.</math>

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 ({{nowrap|2''r'' − ''x''}}) 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 ({{nowrap|2''r'' − ''x'')''x'' {{=}} (''y'' / 2)<sup>2</sup>}}. 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|midpoint]] {{math|'''M'''}} of the diameter.
* Construct the circle with centre {{math|'''M'''}} passing through one of the endpoints of the diameter (it will also pass through the other endpoint).

[[File:Circunferencia 10.svg|thumb|Construct a circle through points A, B and C by finding the perpendicular bisectors (red) of the sides of the triangle (blue). Only two of the three bisectors are needed to find the centre.]]

===Construction through three noncollinear points===
* Name the points {{math|'''P'''}}, {{math|'''Q'''}} and {{math|'''R'''}},
* Construct the perpendicular bisector of the segment {{math|{{overline|'''PQ'''}}}}.
* Construct the perpendicular bisector of the segment {{math|{{overline|'''PR'''}}}}.
* Label the point of intersection of these two perpendicular bisectors {{math|'''M'''}}. (They meet because the points are not collinear).
* Construct the circle with centre {{math|'''M'''}} passing through one of the points {{math|'''P'''}}, {{math|'''Q'''}} or {{math|'''R'''}} (it will also pass through the other two points).

==Circle of Apollonius==
<!--This section is linked from [[Sphere]] among others-->
{{see also|Circles of Apollonius}}
[[Image:Apollonius circle definition labels.svg|thumb|250px|left|Apollonius' definition of a circle: {{nowrap|''d''<sub>1</sub>/''d''<sub>2</sub>}} constant]]
[[Biography:Apollonius of Perga|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''.<ref>{{cite journal | last = Harkness | first = James | title = Introduction to the theory of analytic functions |journal=Nature |volume=59 |issue=1530 |year=1898 |page = 30 | url = https://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01680002 |archive-url=https://web.archive.org/web/20081007134238/http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01680002 | url-status = dead |archive-date=2008-10-07 |bibcode=1899Natur..59..386B |doi=10.1038/059386a0 |s2cid=4030420 }}</ref><ref>Ogilvy, C. Stanley, ''Excursions in Geometry'', Dover, 1969, 14–17.</ref> (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|angle bisector theorem]] the line segment ''PC'' will bisect the interior angle ''APB'', since the segments are similar:
<math display="block">\frac{AP}{BP} = \frac{AC}{BC}.</math>

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<ref>Altshiller-Court, Nathan, ''College Geometry'', Dover, 2007 (orig. 1952).</ref>{{rp|p=15}} 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|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:
<math display="block">\bigl|[A, B; C, P]\bigr| = 1.</math>

Stated another way, ''P'' is a point on the circle of Apollonius if and only if the cross-ratio {{nobr|[''A'', ''B''; ''C'', ''P'']}} is on the unit circle in the complex plane.

==={{anchor|Generalized circles}} Generalised circles===
{{See also|Generalised circle}}
If ''C'' is the midpoint of the segment ''AB'', then the collection of points ''P'' satisfying the Apollonius condition
<math display="block">\frac{|AP|}{|BP|} = \frac{|AC|}{|BC|}</math>
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|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.<ref>[http://mathworld.wolfram.com/Incircle.html Incircle – from Wolfram MathWorld] {{webarchive|url=https://web.archive.org/web/20120121111333/http://mathworld.wolfram.com/Incircle.html |date=2012-01-21 }}. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.</ref>

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]].<ref>[http://mathworld.wolfram.com/Circumcircle.html Circumcircle – from Wolfram MathWorld] {{webarchive|url=https://web.archive.org/web/20120120120814/http://mathworld.wolfram.com/Circumcircle.html |date=2012-01-20 }}. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.</ref>

A [[Tangential polygon|tangential polygon]], such as a [[Tangential quadrilateral|tangential quadrilateral]], is any [[Convex polygon|convex polygon]] within which a circle can be inscribed that is tangent to each side of the polygon.<ref>[http://mathworld.wolfram.com/TangentialPolygon.html Tangential Polygon – from Wolfram MathWorld] {{webarchive|url=https://web.archive.org/web/20130903051014/http://mathworld.wolfram.com/TangentialPolygon.html |date=2013-09-03 }}. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.</ref> Every [[Regular polygon|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|bicentric polygon]].

A [[Hypocycloid|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 various other figures:
* The series of [[Regular polygon|regular polygon]]s with ''n'' sides has the circle as its limit as ''n'' approaches infinity. This fact was applied by [[Biography:Archimedes|Archimedes]] to [[Approximations of π|approximate π]].
* 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|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|superellipse]] has an equation of the form <math>\left|\frac{x}{a}\right|^n\! + \left|\frac{y}{b}\right|^n\! = 1</math> for positive ''a'', ''b'', and ''n''. A supercircle has {{nowrap|1=''b'' = ''a''}}. A circle is the special case of a supercircle in which {{nowrap|''n'' {{=}} 2}}.
* 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|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.

==Locus of constant sum==
Consider a finite set of <math>n</math> 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 centre is at the centroid of the given points.<ref>{{cite journal | last1 = Apostol | first1 = Tom | last2 = Mnatsakanian | first2 = Mamikon | date = 2003 | title = Sums of squares of distances in m-space | journal= American Mathematical Monthly | volume=110 | issue=6 | pages = 516–526 | doi = 10.1080/00029890.2003.11919989 | s2cid = 12641658}}</ref>
A generalisation for higher powers of distances is obtained if, instead of <math>n</math> points, the vertices of the regular polygon <math>P_n</math> are taken.<ref name="Mamuka">{{cite journal |last1=Meskhishvili |first1=Mamuka |date=2020 |title=Cyclic Averages of Regular Polygons and Platonic Solids |url=https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065 |journal=Communications in Mathematics and Applications |volume=11 |pages=335–355 |doi=10.26713/cma.v11i3.1420 |doi-broken-date=11 July 2025 |arxiv=2010.12340 |access-date=17 May 2021 |archive-date=22 April 2021 |archive-url=https://web.archive.org/web/20210422211229/https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065 |url-status=live }}</ref> The locus of points such that the sum of the <math>2m</math>-th power of distances <math>d_i</math> to the vertices of a given regular polygon with circumradius <math>R</math> is constant is a circle, if
<math display="block">\sum_{i=1}^n d_i^{2m} > nR^{2m} , \quad \text{  where } ~ m = 1, 2, \dots, n-1;</math>
whose centre is the centroid of the <math>P_n</math>.

In the case of the [[Equilateral triangle|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==
{{main|Squaring the circle}}
Squaring the circle is the problem, proposed by [[History: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 ({{pi}}) is a [[Transcendental number|transcendental number]], rather than an [[Algebraic number|algebraic irrational number]]; that is, it is not the [[Root of a function|root]] of any [[Polynomial|polynomial]] with [[Rational number|rational]] coefficients. Despite the impossibility, this topic continues to be of interest for pseudomath enthusiasts.

== Generalisations ==
===In other ''p''-norms===
[[Image:Vector-p-Norms qtl1.svg|thumb|right|Illustrations of unit circles (see also [[Superellipse|superellipse]]) in different {{math|''p''}}-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding {{math|''p''}}).]]
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
<math display="block"> \left\| x \right\| _p = \left( \left|x_1\right|^p + \left|x_2\right|^p + \dotsb + \left|x_n\right|^p \right) ^{1/p} .</math>
In Euclidean geometry, ''p'' = 2, giving the familiar
<math display="block"> \left\| x \right\| _2 = \sqrt{ \left|x_1\right|^2 + \left|x_2\right|^2 + \dotsb + \left|x_n\right|^2 } .</math>

In [[Taxicab geometry|taxicab geometry]], ''p'' = 1. Taxicab circles are [[Square|square]]s with sides oriented at a 45° angle to the coordinate axes. While each side would have length <math>\sqrt{2} r</math> 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 <math>\pi </math> is 4 in this geometry. The formula for the unit circle in taxicab geometry is <math>|x| + |y| = 1</math> in Cartesian coordinates and
<math display="block">r = \frac{1}{\left| \sin \theta\right| + \left|\cos\theta\right|}</math>
in polar coordinates.

A circle of radius 1 (using this distance) is the [[Von Neumann neighborhood|von Neumann neighborhood]] of its centre.

A circle of radius ''r'' for the [[Chebyshev distance]] ([[Lp space|''L''<sub>∞</sub> 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''<sub>1</sub> and ''L''<sub>∞</sub> metrics does not generalise to higher dimensions.

===Topological definition===
The circle is the one-dimensional [[Hypersphere|hypersphere]] (the 1-sphere).

In [[Topology|topology]], a circle is not limited to the geometric concept, but to all of its [[Homeomorphism|homeomorphism]]s. Two topological circles are equivalent if one can be transformed into the other via a deformation of [[Real coordinate space|'''R'''<sup>3</sup>]] upon itself (known as an [[Ambient isotopy|ambient isotopy]]).<ref name="gamelin">{{cite book | last = Gamelin | first = Theodore | title = Introduction to topology | url = https://archive.org/details/introductiontoto00game | url-access = registration | publisher = Dover Publications | location = Mineola, N.Y | year = 1999 | isbn = 0486406806 }}</ref>

===Orientation===
An '''oriented circle''' is an ordinary circle with an [[Orientation (vector space)|orientation]] represented by a binary value attached to it, which is either <math>1</math> or <math>-1</math>. The only exception is a circle of radius zero, which has orientation equal to <math>0</math>. A point is equivalent to an oriented circle of radius zero. If an oriented circle has orientation equal to <math>1</math>, then the circle is said to be "anti-clockwise" oriented; if it has orientation equal to <math>-1</math> then it is "[[Physics:Clockwise|clockwise]]" oriented. The (signed) radius of an oriented circle is defined to be the radius <math>r</math> of the underlying unoriented circle multiplied by the orientation. A circle orientation is assumed in the measure of signed angles.

==Specially named circles==
{{col-begin}}
{{col-break}}
* [[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]]

{{col-break}}
===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)]]
* [[Sine-triple-angle circle]]
* [[Spieker circle]]
* Van Lamoen circle

{{col-break}}

===Of certain quadrilaterals===
* Eight-point circle of an orthodiagonal quadrilateral

===Of a conic section===
* [[Director circle]]
* Directrix circle

===Of a torus===
* [[Villarceau circles]]
{{col-end}}

==See also==

* {{Annotated link |Affine sphere}}
* {{Annotated link |Apeirogon}}
* {{Annotated link |Circle fitting}}
* {{Annotated link |Distance}}
* {{Annotated link |Gauss circle problem}}
* {{Annotated link |Inversion in a circle}}
* {{Annotated link |Line–circle intersection}}
* {{Annotated link |List of circle topics}}
* {{Annotated link |Sphere}}
* {{Annotated link |Three points determine a circle}}
* {{Annotated link |Translation of axes}}

==Notes==
{{Notelist}}

==References==
{{Reflist}}

==Further reading==
* {{cite book |author=Pedoe, Dan |title=Geometry: a comprehensive course |url=https://archive.org/details/geometrycomprehe0000pedo |url-access=registration |publisher=Dover |year=1988|isbn=9780486658124 }}

==External links==
{{Commons and category|Circles|Circles}}
{{EB1911 poster|Circle}}
* {{springer|title=Circle|id=p/c022260}}
* {{PlanetMath|urlname=circle |title=Circle}}
* {{MathWorld |urlname=Circle |title=Circle}}
* {{cite web| url = http://www.mathopenref.com/tocs/circlestoc.html | title = Interactive Java applets | quote = for the properties of and elementary constructions involving circles}}
* {{cite web| url = http://www.mathwarehouse.com/geometry/circle/interactive-circle-equation.php | title = Interactive Standard Form Equation of Circle | quote = Click and drag points to see standard form equation in action}}
* {{cite web| url = http://www.cut-the-knot.org/pythagoras/Munching/circle.shtml | title = Munching on Circles | publisher = Cut-the-Knot}}


[[Category:Circles| ]]
[[Category:Elementary shapes]]
[[Category:Conic sections]]

{{Sourceattribution|Circle}}