# Circle

Last updated

Circle
A circle
circumference C
diameter D
radius R
centre or origin O
Type Conic section
Symmetry group O(2)
Area πR2
Perimeter C = 2πR

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, thecentre. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with ${\displaystyle r=0}$ (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

## Contents

Specifically, a circle is a simple closed curve 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 .[ citation needed ]

A circle may also be defined as a special kind of ellipse in which the two foci 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.

## Euclid's definition

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.

Euclid, Elements , Book I [1] :4

## Topological definition

In the field of topology, a circle is not limited to the geometric concept, but to all of its homeomorphisms. Two topological circles are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy). [2]

## Terminology

• Annulus: a ring-shaped object, the region bounded by two concentric circles.
• Arc: any connected part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle.
• Centre: the point equidistant from all points on the circle.
• Chord: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
• Circumference: the length of one circuit along the circle, or the distance around the circle.
• Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
• Disc: the region of the plane bounded by a circle.
• 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: a region bounded by two radii of equal length with a common centre and either of the two possible arcs, determined by this centre and the endpoints of the radii.
• Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the centre of the circle to which their arc belongs 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 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.

 Chord, secant, tangent, radius, and diameter Arc, sector, and segment

## History

The word circle derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself a metathesis of the Homeric Greek κρίκος (krikos), meaning "hoop" or "ring". [3] The origins of the words circus 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, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus.

Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles. [4] [5]

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 256/81 (3.16049...) as an approximate value of π. [6]
• 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles.
• In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation.
• 1880 CE – Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle. [7]

## Analytic results

### Circumference

The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the circumference C is related to the radius r and diameter d by:

${\displaystyle C=2\pi r=\pi d.\,}$

### Area enclosed

As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, [8] which comes to π multiplied by the radius squared:

${\displaystyle \mathrm {Area} =\pi r^{2}.\,}$

Equivalently, denoting diameter by d,

${\displaystyle \mathrm {Area} ={\frac {\pi d^{2}}{4}}\approx 0{.}7854d^{2},}$

that is, approximately 79% of the circumscribing square (whose side is of length d).

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.

### Equations

#### Cartesian coordinates

Equation of a circle

In an xy Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that

${\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}.}$

This equation, known as the equation of the circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |xa| and |yb|. If the circle is centred at the origin (0, 0), then the equation simplifies to

${\displaystyle x^{2}+y^{2}=r^{2}.}$
Parametric form

The equation can be written in parametric form using the trigonometric functions sine and cosine as

${\displaystyle x=a+r\,\cos t,}$
${\displaystyle y=b+r\,\sin t,}$

where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the positive x axis.

An alternative parametrisation of the circle is

${\displaystyle x=a+r{\frac {1-t^{2}}{1+t^{2}}},}$
${\displaystyle y=b+r{\frac {2t}{1+t^{2}}}.}$

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

3-point form

The equation of the circle determined by three points ${\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})}$ not on a line is obtained by a conversion of the 3-point form of a circle equation:

${\displaystyle {\frac {({\color {green}x}-x_{1})({\color {green}x}-x_{2})+({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {green}x}-x_{2})-({\color {red}y}-y_{2})({\color {green}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.}$
Homogeneous form

In homogeneous coordinates, each conic section with the equation of a circle has the form

${\displaystyle x^{2}+y^{2}-2axz-2byz+cz^{2}=0.}$

It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: i: 0). These points are called the circular points at infinity.

#### Polar coordinates

In polar coordinates, the equation of a circle is[ vague ]

${\displaystyle r^{2}-2rr_{0}\cos(\theta -\phi )+r_{0}^{2}=a^{2},}$

where a is the radius of the circle, ${\displaystyle (r,\theta )}$ are the polar coordinates of a generic point on the circle, and ${\displaystyle (r_{0},\phi )}$ are the polar coordinates of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x axis to the line connecting the origin to the centre of the circle). For a circle centred on the origin, i.e. r0 = 0, this reduces to r = a. When r0 = a, or when the origin lies on the circle, the equation becomes

${\displaystyle r=2a\cos(\theta -\phi ).}$

In the general case, the equation can be solved for r, giving

${\displaystyle r=r_{0}\cos(\theta -\phi )\pm {\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\theta -\phi )}}.}$

Note that without the ± sign, the equation would in some cases describe only half a circle.

#### Complex plane

In the complex plane, a circle with a centre at c and radius r has the equation

${\displaystyle |z-c|=r.}$

In parametric form, this can be written as

${\displaystyle z=re^{it}+c.}$

The slightly generalised equation

${\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q}$

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

### Tangent lines

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1a)x + (y1b)y = c. Evaluating at (x1, y1) determines the value of c, and the result is that the equation of the tangent is

${\displaystyle (x_{1}-a)x+(y_{1}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{1},}$

or

${\displaystyle (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.}$

If y1b, then the slope of this line is

${\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}-a}{y_{1}-b}}.}$

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

${\displaystyle x_{1}x+y_{1}y=r^{2},}$

and its slope is

${\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}}{y_{1}}}.}$

## 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, and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.
• All circles are similar.
• A circle circumference and radius are proportional.
• The area enclosed and the square of its radius are proportional.
• The constants of proportionality are 2π and π respectively.
• The circle that is centred at the origin with radius 1 is called the unit 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 through the centre bisecting a chord is perpendicular to the chord.
• If a central angle and an 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.
• 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 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 a2 + b2 + c2 + d2 equals the square of the diameter. [9]
• 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 8r2 − 4p2, where r is the circle radius, and p is the distance from the centre point to the point of intersection. [10]
• 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. [11] :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 DAQ = 1/2arc(AQ).

### Theorems

• The chord theorem states that if two chords, CD and EB, intersect at A, then AC × AD = AB × AE.
• If two secants, AE and AD, also cut the circle at B and C respectively, then AC × AD = AB × AE (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 AF2 = 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°, then = r √2, where 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 (${\displaystyle {\overset {\frown }{DE}}}$ and ${\displaystyle {\overset {\frown }{BC}}}$). That is, ${\displaystyle 2\angle {CAB}=\angle {DOE}-\angle {BOC}}$, where O is the centre of the circle (secant–secant theorem).

### Inscribed angles

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

### Sagitta

The sagitta (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:

${\displaystyle r={\frac {y^{2}}{8x}}+{\frac {x}{2}}.}$

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 (2rx) 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 (2rx)x = (y / 2)2. 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 on the centre point, the movable leg on the point on the circle and rotate the compass.

### Construction with given diameter

• Construct the midpoint M of the diameter.
• Construct the circle with centre M 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 P, Q and R,
• Construct the perpendicular bisector of the segment PQ.
• Construct the perpendicular bisector of the segment PR.
• Label the point of intersection of these two perpendicular bisectors M. (They meet because the points are not collinear).
• Construct the circle with centre M passing through one of the points P, Q or R (it will also pass through the other two points).

## Circle of Apollonius

Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ratio (other than 1) of distances to two fixed foci, A and B. [12] [13] (The set of points where the distances are equal is the perpendicular bisector of segment AB, a line.) That circle is sometimes said to be drawn about two points.

The proof is in two parts. First, one must prove that, given two foci A and B and a ratio of distances, any point P satisfying the ratio of distances must fall on a particular circle. Let C be another point, also satisfying the ratio and lying on segment AB. By the angle bisector theorem the line segment PC will bisect the interior angle APB, since the segments are similar:

${\displaystyle {\frac {AP}{BP}}={\frac {AC}{BC}}.}$

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 [14] :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 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:

${\displaystyle {\big |}[A,B;C,P]{\big |}=1.}$

Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio [A, B; C, P] 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

${\displaystyle {\frac {|AP|}{|BP|}}={\frac {|AC|}{|BC|}}}$

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 unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle. [15]

About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices. [16]

A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon. [17] 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.

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 has an equation of the form ${\displaystyle \left|{\frac {x}{a}}\right|^{n}\!+\left|{\frac {y}{b}}\right|^{n}\!=1}$ for positive a, b, and n. A supercircle has b = a. A circle is the special case of a supercircle in which 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 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

${\displaystyle \left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.}$

In Euclidean geometry, p = 2, giving the familiar

${\displaystyle \left\|x\right\|_{2}={\sqrt {|x_{1}|^{2}+|x_{2}|^{2}+\dotsb +|x_{n}|^{2}}}.}$

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 ${\displaystyle {\sqrt {2}}r}$ using a Euclidean metric, where r is the circle's radius, its length in taxicab geometry is 2r. Thus, a circle's circumference is 8r. Thus, the value of a geometric analog to ${\displaystyle \pi }$ is 4 in this geometry. The formula for the unit circle in taxicab geometry is ${\displaystyle |x|+|y|=1}$ in Cartesian coordinates and

${\displaystyle r={\frac {1}{|\sin \theta |+|\cos \theta |}}}$

in polar coordinates.

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

A circle of radius r for the Chebyshev distance (L metric) on a plane is also a square with side length 2r parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between L1 and L metrics does not generalize to higher dimensions.

## Locus of constant sum

Consider a finite set of ${\displaystyle n}$ points in the plane. The locus of points such that the sum of the squares of the distances to the given points is constant is a circle, whose centre is at the centroid of the given points. [18] A generalization for higher powers of distances is obtained if under ${\displaystyle n}$ points the vertices of the regular polygon ${\displaystyle P_{n}}$ are taken. [19] The locus of points such that the sum of the ${\displaystyle (2m)}$-th power of distances ${\displaystyle d_{i}}$ to the vertices of a given regular polygon with circumradius ${\displaystyle R}$ is constant is a circle, if

${\displaystyle \sum _{i=1}^{n}d_{i}^{2m}>nR^{2m}}$, where ${\displaystyle m}$=1,2,…, ${\displaystyle n}$-1;

whose centre is the centroid of the ${\displaystyle P_{n}}$.

In the case of the equilateral triangle, the loci of the constant sums of the second and fourth powers are circles, whereas for the square, the loci are circles for the constant sums of the second, fourth, and sixth powers. For the regular pentagon the constant sum of the eighth powers of the distances will be added and so forth.

## Squaring the circle

Squaring the circle is the problem, proposed by 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 (π) is a transcendental number, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. Despite the impossibility, this topic continues to be of interest for pseudomath enthusiasts.

## Significance in art and symbolism

From the time of the earliest known civilisations – such as the Assyrians and ancient Egyptians, those in the Indus Valley and along the Yellow River in China, and the Western civilisations of ancient Greece and Rome during classical Antiquity – the circle has been used directly or indirectly in visual art to convey the artist's message and to express certain ideas. However, differences in worldview (beliefs and culture) had a great impact on artists’ perceptions. While some emphasised the circle's perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of existence, but in religious traditions it represents heavenly bodies and divine spirits. The circle signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example, a compass, a halo, the vesica piscis and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, the Dharma wheel, a rainbow, mandalas, rose windows and so forth. [20]

## Unicode

Unicode provides various circle symbol: [21]

SymbolNameCode Point
LARGE CIRCLEU+25EF
IDEOGRAPHIC NUMBER ZEROU+3007
FULLWIDTH LATIN CAPITAL LETTER OU+FF2F
COMBINING ENCLOSING CIRCLEU+20DD
HEAVY LARGE CIRCLEU+2B55
MEDIUM BLACK CIRCLEU+26AB
🔴LARGE RED CIRCLEU+1F534
OLATIN CAPITAL LETTER OU+004F
🔵LARGE BLUE CIRCLEU+1F535
CIRCLED PLUSU+2295
CIRCLED MINUSU+2296
CIRCLED DIVISION SLASHU+2298
CIRCLED RING OPERATORU+229A
CIRCLED ASTERISK OPERATORU+229B
CIRCLED EQUALSU+229C
CIRCLED DASHU+229D
SHADOWED WHITE CIRCLEU+274D
⦿CIRCLED BULLETU+29BF
🔘RADIO BUTTONU+1F518
CIRCLED OPEN CENTRE EIGHT POINTED STARU+2742
RADIOACTIVE SIGNU+2622
CIRCLED TIMESU+2297
CIRCLED DOT OPERATORU+2299
CLOCKWISE CLOSED CIRCLE ARROWU+2941
ANTICLOCKWISE CLOSED CIRCLE ARROWU+2940
CIRCLED POSTAL MARKU+3036
🎯DIRECT HITU+1F3AF
CIRCLED DIVISION SIGNU+2A38
MULTIPLICATION SIGN IN DOUBLE CIRCLEU+2A37
PARTIALLY-RECYCLED PAPER SYMBOLU+267D
RECYCLED PAPER SYMBOLU+267C
YIN YANGU+262F
PEACE SYMBOLU+262E
TIBETAN SYMBOL NOR BU NYIS -KHYILU+0FCA
FISHEYEU+25C9
WHITE CIRCLEU+25CB
DOTTED CIRCLEU+25CC
CIRCLE WITH VERTICAL FILLU+25CD
BULLSEYEU+25CE
BLACK CIRCLEU+25CF
CIRCLE WITH LEFT HALF BLACKU+25D0
CIRCLE WITH RIGHT HALF BLACKU+25D1
CIRCLE WITH LOWER HALF BLACKU+25D2
CIRCLE WITH UPPER HALF BLACKU+25D3
CIRCLE WITH UPPER RIGHT QUADRANT BLACKU+25D4
CIRCLE WITH ALL BUT UPPER LEFT QUADRANT BLACKU+25D5
LEFT HALF BLACK CIRCLEU+25D6
RIGHT HALF BLACK CIRCLEU+25D7
INVERSE BULLETU+25D8
INVERSE WHITE CIRCLEU+25D9
UPPER HALF INVERSE WHITE CIRCLEU+25DA
LOWER HALF INVERSE WHITE CIRCLEU+25DB
UPPER LEFT QUADRANT CIRCULAR ARCU+25DC
UPPER RIGHT QUADRANT CIRCULAR ARCU+25DD
LOWER RIGHT QUADRANT CIRCULAR ARCU+25DE
LOWER LEFT QUADRANT CIRCULAR ARCU+25DF
UPPER HALF CIRCLEU+25E0
LOWER HALF CIRCLEU+25E1
BLACK LARGE CIRCLEU+2B24
WHITE BULLETU+25E6
EMPTY SETU+2205
RING OPERATORU+2218
ERROR-BARRED BLACK CIRCLEU+29F3
ERROR-BARRED WHITE CIRCLEU+29F2
BLACK CIRCLE WITH DOWN ARROWU+29ED
WHITE CIRCLE WITH DOWN ARROWU+29EC
CIRCLE WITH TWO HORIZONTAL STROKES TO THE RIGHTU+29C3
CIRCLE WITH SMALL CIRCLE TO THE RIGHTU+29C2
CIRCLED GREATER-THANU+29C1
CIRCLED LESS-THANU+29C0
UP ARROW THROUGH CIRCLEU+29BD
CIRCLED ANTICLOCKWISE-ROTATED DIVISION SIGNU+29BC
CIRCLE DIVIDED BY HORIZONTAL BAR AND TOP HALF DIVIDED BY VERTICAL BARU+29BA
CIRCLED PERPENDICULARU+29B9
TELEPHONE LOCATION SIGNU+2706
NO ENTRYU+26D4
BASEBALLU+26BE
UNMARRIED PARTNERSHIP SYMBOLU+26AF
DIVORCE SYMBOLU+26AE
MARRIAGE SYMBOLU+26AD
BLACK CIRCLE WITH TWO WHITE DOTSU+2689
BLACK CIRCLE WITH WHITE DOT RIGHTU+2688
WHITE CIRCLE WITH TWO DOTSU+2687
WHITE CIRCLE WITH DOT RIGHTU+2686
PERMANENT PAPER SIGNU+267E

## Related Research Articles

In mathematics, an ellipse is a plane curve surrounding two 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 type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

A sphere is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

In elementary geometry, two geometric objects are perpendicular if their intersection forms right angles at the point of intersection called a foot. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂. Perpendicular intersections can happen between two lines, between a line and a plane, and between two planes.

In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842-3) and Kelvin (1845).

In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.

In geometry, a set of points are said to be concyclic if they lie on a common circle. All concyclic points are equidistant from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, but four or more such points in the plane are not necessarily concyclic.

In mathematics, a pedal curve of a given curve results from the orthogonal projection of a fixed point on the tangent lines of this curve. More precisely, for a plane curve C and a given fixed pedal pointP, the pedal curve of C is the locus of points X so that the line PX is perpendicular to a tangent T to the curve passing through the point X. Conversely, at any point R on the curve C, let T be the tangent line at that point R; then there is a unique point X on the tangent T which forms with the pedal point P a line perpendicular to the tangent T – the pedal curve is the set of such points X, called the foot of the perpendicular to the tangent T from the fixed point P, as the variable point R ranges over the curve C.

In hyperbolic geometry, two lines are said to be ultraparallel if they do not intersect and are not limiting parallel.

In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.

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A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the centre of the circle that is less than π radians ; and the other arc, the major arc, subtends an angle greater than π radians. The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a chord of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.

In Euclidean geometry, a tangential 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 incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.

In geometry, a homothetic center is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another. If the center is external, the two figures are directly similar to one another; their angles have the same rotational sense. If the center is internal, the two figures are scaled mirror images of one another; their angles have the opposite sense.

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In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also rarely been called a double circle quadrilateral and double scribed quadrilateral.

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 (perpendicular) to each other.

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.

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