circle

Dictionary: cir·cle (sûr'kəl) pronunciation

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circle

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A plane curve everywhere equidistant from a given fixed point, the center.
A planar region bounded by a circle.
Something, such as a ring, shaped like such a plane curve.
A circular course, circuit, or orbit: a satellite's circle around the earth.
A traffic circle.
A curved section or tier of seats in a theater.
A series or process that finishes at its starting point or continuously repeats itself; a cycle.
A group of people sharing an interest, activity, or achievement: well-known in artistic circles.
A territorial or administrative division, especially of a province, in some European countries.
A sphere of influence or interest; domain.
Logic. A vicious circle.

v., -cled, -cling, -cles.

v.tr.

To make or form a circle around; enclose. See synonyms at surround.
To move in a circle around.

v.intr.

To move in a circle. See synonyms at turn.

idiom:

circle the wagons

To take a defensive position; become defensive.

[Middle English cercle, from Old French, from Latin circulus, diminutive of circus, circle, from Greek kirkos, krikos.]

circler cir'cler (-klər) n.

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5min Related Video: circle

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Sci-Tech Encyclopedia: Circle

The curve that is the locus of points in a plane with equal distance (radius) from a fixed point (center). In elementary mathematics, circle often refers to the finite portion of the plane bounded by a curve (circumference) all points of which are equidistant from a fixed point of the plane, that is, a circular disk. Circles are conic sections and are defined analytically by certain second-degree equations in cartesian coordinates. The ancient Greeks formulated the problem of “squaring the circle,” that is, to construct, with compasses and unmarked straightedge only, a square whose area is equal to that of a given circle. It was not until 1882 that this was shown to be impossible, when F. Lindemann proved that the ratio of the length of a circle to its diameter (denoted by π) is not the root of any algebraic equation with integer coefficients. Electronic computers have calculated π to over 10¹² decimal places.

The area of a circle (circular disk) with radius r is πr²; the length (circumference) is 2πr. The area enclosed by a circle is greater than that bounded by any other curve of the same length. See also Analytic geometry; Conic section.

Financial & Investment Dictionary: Circle

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Underwriter's way of designating potential purchasers and amounts of a securities issue during the Registration period, before selling is permitted. Registered representatives canvass prospective buyers and report any interest to the underwriters, who then circle the names on their list.

Thesaurus: circle

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noun

A closed plane curve everywhere equidistant from a fixed point or something shaped like this: band, circuit, disk, gyre, ring, wheel. Archaic orb. See geometry.
A course, process, or journey that ends where it began or repeats itself: circuit, cycle, orbit, round, tour, turn. See repetition.
A group of people sharing an interest, activity, or achievement: crowd, group, set. See group.
A particular social group: clique, coterie, crowd, set. Informal bunch, gang. See group.
A sphere of activity, experience, study, or interest: area, arena, bailiwick, department, domain, field, orbit, province, realm, scene, subject, terrain, territory, world. Slang bag. See territory.

verb

To shut in on all sides: begird, beset, compass, encircle, encompass, environ, gird, girdle, hedge, hem, ring, surround. See open/close.
To move or cause to move in circles or around an axis: circumvolve, gyrate, orbit, revolve, rotate, turn, wheel. See move/halt, repetition.

Music Encyclopedia: Circles

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Work by Berio for female voice, harp and two percussionists (1960), settings of poems by E.E. Cummings; the singer moves to different positions on the platform.

Britannica Concise Encyclopedia: circle

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Geometrical curve, one of the conic sections, consisting of the set of all points the same distance (the radius) from a given point (the centre). A line connecting any two points on a circle is called a chord, and a chord passing through the centre is called a diameter. The distance around a circle (the circumference) equals the length of a diameter multiplied by p (see pi). The area of a circle is the square of the radius multiplied by p. An arc consists of any part of a circle encompassed by an angle with its vertex at the centre (central angle). Its length is in the same proportion to the circumference as the central angle is to a full revolution.

For more information on circle, visit Britannica.com.

English Folklore: circles

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circling

Symbolically, a circle can stand for perfection, wholeness, or a boundary (protective or confining); circling round something can be a way of honouring or blessing it, or, conversely, of receiving blessing or power from it. Circling can also summon a supernatural being—it is one of the commonest English local traditions that if you run round a specified mound, tree, cross, grave, church, or stone at a specified time and/or a specified number of times without stopping, you will raise a ghost, or the Devil; the condition is less easy than it seems, since running round a small object causes giddiness, and round a large one is exhausting. The circle as boundary is exemplified by the common instruction in manuals of magic to draw a circle round oneself as protection against spirits summoned, or to conjure the spirit into a circle which will confine it; more prosaically, it appears also in the Devonshire belief that a snake cannot escape a circle drawn round it with an ash stick (Bray, 1838: 95).

See also LEFTWARD and RIGHTWARD MOVEMENT.

Columbia Encyclopedia: circle

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circle, closed plane curve consisting of all points at a given distance from some fixed point, called the center. A circle is a conic section cut by a plane perpendicular to the axis of the cone. The term circle is also used to refer to the region enclosed by the curve, more properly called a circular region. The radius of a circle is any line segment connecting the center and a point on the curve; the term is also used for the length r of this segment, i.e., the common distance of all points on the curve from the center. Similarly, the circumference of a circle is either the curve itself or its length of arc. A line segment whose two ends lie on the circumference is a chord; a chord through the center is the diameter. A secant is a line of indefinite length intersecting the circle at two points, the segment of it within the circle being a chord. A tangent to a circle is a straight line touching the circle at only one point, the point of contact, or tangency, and is always perpendicular to the radius drawn to this point. A circle is inscribed in a polygon if each side of the polygon is tangent to the circle; a circle is circumscribed about a polygon if all the vertices of the polygon lie on the circumference. The length of the circumference C of a circle is equal to π (see pi) times twice the radius distance r, or C=2πr. The area A bounded by a circle is given by A=πr². Greek geometry left many unsolved problems about circles, including the problem of squaring the circle, i.e., constructing a square with an area equal to that of a given circle, using only a straight edge and compass; it was finally proved impossible in the late 19th cent. (see geometric problems of antiquity). In modern mathematics the circle is the basis for such theories as inversive geometry and certain non-Euclidean geometries. The circle figures significantly in many cultures. In religion and art it frequently symbolizes heaven, eternity, or the universe.

Occultism & Parapsychology Encyclopedia: Circles

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The circle is the most common space created for the working of magic and witchcraft. It stands in sharp contrast to the rectangular space that the average Christian church defines. The circle is easily drawn on the ground and just as easily erased. The circle has been a popular form for worship since ancient times as demonstrated by numerous stone monuments found around the world.

In modern magical and Wiccan practice, the circle is seen as both a protective barrier and a container of energy. It is the visible manifestation of a sphere that completely surrounds the worker of magic. Where the invisible sphere intersects the ground or floor, a circle is defined. While occasionally a more permanent circle is drawn and remains for regular workings, the circle is usually created only at the beginning of a magical ritual and dissolved at its close.

Modern magical rituals begin with the imaginal setting of a sphere of energy around the individual or group performing the ritual. Commonly, there are specific words that are spoken to create the sphere or circle. Most Wiccans believe in the existence of an array of spirit beings, from deities to elemental spirits. Most rituals are designed to invoke one or more of these deities and the intrusion of unwanted entities would disturb the focus of the ritual. In such settings, the circle is seen as a barrier that protects the ritual and keeps entities attracted by the power raised by the ritual from disturbing its fruitful conclusion. The ritual is closed with a banishing act dispersing any attending entities.

Modern rituals are also seen as acts that raise, focus, and direct energy to a specific purpose such as the healing of someone or the gaining of some particular favor. In such thinking, the sphere or circle is seen as a container that holds the energy so raised until the ritual's climax, when it is sent forth to do its work.

In modern Neo-Paganism, where worship predominates over magic, the idea of creating the circles as the creation of sacred space, apart from the mundane world, predominates. Sacred space is, or becomes, space in which the veil dividing the common everyday world from the realm of spirits is thin and communication is possible. While some sacred space is defined by the environment, a particularly beautiful or striking spot, it can be created anywhere. In the pantheistic Pagan world, all space is ultimately seen as sacred. Sacred space is often entered only after participants have cleansed themselves and donned special dress, commonly a ritual robe, or as in the case of some Wiccan groups, in the nude.

Sources:

Adler, Margot. Drawing Down the Moon. Boston: Beacon Press, 1979.

Crowley, Vivianne. Principles of Paganism. London: Thor-sons, 1996.

Veterinary Dictionary: circling

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Persistent walking in circles; it may be caused by deviation of the head because of a bend in the neck, or be due to rotation of the head, e.g. caused by listeriosis or brain abscess. A sign of unilateral vestibular disease or cerebral lesion (toward the side of the lesion; adverse syndrome).

c. disease — see listeriosis.

Word Tutor: circle

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IN BRIEF: A closed curved line forming a perfectly round, flat figure.

The art project started with the drawing of a circle in the center of the paper.

Tutor's tip: The mice formed a "circle" (a curved line, every point of which is equally distant from the center) around the "cereal" (any grain used for food).

Dream Symbol: Circle

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A circle encompasses many meanings in numerous areas: the wholeness of numbers in mathematics, the spiritual oneness depicted by the circle and the mandala, protection from evil by the ritual drawing of a circle, bringing attention to something by circling it. It may also express frustrations, as when one doodles in circles or goes around in circles. Socially, it may represent being "in" the right circle of friends. The love relationship is sometimes symbolized by the wearing of a ring, around the finger, the neck, or in the nose. In Jungian psychology the circle is a symbol of the self archetype. (See also Zero)

Wikipedia: Circle

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This article is about the shape and mathematical concept of circle. For list of topics related to circles, see List of circle topics. For all other uses, see Circle (disambiguation).

Circle illustration showing a radius, a diameter, the center and the circumference

Tycho crater, one of many examples of circles that arise in nature. NASA photo

A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are the same distance from a given point called the centre. The common distance of the points of a circle from its center is called its radius.

Circles are simple closed curves which divide 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 (known as the perimeter) or to the whole figure including its interior. However, in strict technical usage, "circle" refers to the perimeter while the interior of the circle is called a disk. The circumference of a circle is the perimeter of the circle (especially when referring to its length).

A circle is a special ellipse in which the two foci are coincident. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

Further terminology

Chord, secant, tangent, and diameter.

Arc, sector, and segment

The diameter of a circle is the length of a line segment whose endpoints lie on the circle and which passes through the centre of the circle. This is the largest distance between any two points on the circle. The diameter of a circle is twice its radius.

The term " radius" can also refer to a line segment from the centre of a circle to its perimeter, and similarly the term "diameter" can refer to a line segment between two points on the perimeter which passes through the centre. In this sense, the midpoint of a diameter is the centre and so it is composed of two radii.

A chord of a circle is a line segment whose two endpoints lie on the circle. The diameter, passing through the circle's centre, is the largest chord in a circle. A tangent to a circle is a straight line that touches the circle at a single point. A secant is an extended chord: a straight line cutting the circle at two points.

An arc of a circle is any connected part of the circle's circumference. A sector is a region bounded by two radii and an arc lying between the radii, and a segment is a region bounded by a chord and an arc lying between the chord's endpoints.

History

The compass in this 13th century manuscript is a symbol of God's act of Creation. Notice also the circular shape of the halo

The circle has been known since before the beginning of recorded history. It is the basis for the wheel, which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry 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.^{[citation needed]}

Some highlights in the history of the circle are:

1700 BC – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of $π$ .^[1]
300 BC – Book 3 of Euclid's Elements deals with the properties of circles.
1880 – Lindemann proves that $π$ is transcendental, effectively settling the millennia-old problem of squaring the circle.^[2]

Analytic results

Length of circumference

For more details on this topic, see Pi.

The ratio of a circle's circumference to its diameter is π (pi), a constant that takes the same value (approximately 3.1416) for all circles. Thus the length of the circumference (c) is related to the radius (r) by

$c = 2 \pi r\,$

or equivalently to the diameter (d) by

$c = \pi d.\,$

Area enclosed

Area of the circle = π × area of the shaded square

Main article: Area of a disk

The area enclosed by a circle is $π$ multiplied by the radius squared:

$Area = \pi r^2.\,$

Equivalently, denoting diameter by d,

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

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.

Equation

Circle of radius r = 1, center (a, b) = (1.2, -0.5)

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

$\left(x - a \right)^2 + \left( y - b \right)^2=r^2.$

This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, 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 centered at the origin (0, 0), then the equation simplifies to

$x^2 + y^2 = r^2. \!\$

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

$x = a+r\,\cos t,\,\!$

$y = b+r\,\sin t\,\!$

where t is a parametric variable, interpreted geometrically as the angle that the ray from the origin to (x, y) makes with the x-axis. Alternatively, in stereographic coordinates, the circle has a parametrization

$x = a + r \frac{1-t^2}{1+t^2}$

$y = b + r \frac{2t}{1+t^2}.$

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

$\ ax^2+ay^2+2b_1xz+2b_2yz+cz^2 = 0.$

It can be proven that a conic section is a circle if and only if the point I(1: i: 0) and J(1: −i: 0) lie on the conic section. These points are called the circular points at infinity.

In polar coordinates the equation of a circle is

$r^2 - 2 r r_0 \cos(\theta - \varphi) + r_0^2 = a^2\,$

where a is the radius of the circle, 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 at the origin, this reduces to simply r = a.

In the complex plane, a circle with a center at c and radius (r) has the equation $|z-c|^2 = r^2\,$ . Since $|z-c|^2 = z\overline{z}-\overline{c}z-c\overline{z}+c\overline{c}$ , the slightly generalized equation $pz\overline{z} + gz + \overline{gz} = q$ for real p, q and complex g is sometimes called a generalised circle. Not all generalised circles are actually circles: a generalized circle is either a (true) circle or a line.

Tangent lines

Main article: Tangent lines to circles

The tangent line through a point P on a circle is perpendicular to the diameter passing through P. The equation of the tangent line to a circle of radius r centered at the origin at the point (x₁, y₁) is

$xx_1+yy_1=r^2. \!\$

Hence, the slope of a circle at (x₁, y₁) is given by:

$\frac{dy}{dx} = - \frac{x_1}{y_1}.$

More generally, the slope at a point (x, y) on the circle $(x - a) 2 + (y - b) 2 = r 2$ , i.e., the circle centered at (a, b) with radius r units, is given by

$\frac{dy}{dx} = \frac{a-x}{y-b},$

provided that $y \neq b$ .

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 center forms a line of reflection symmetry and it has rotational symmetry around the center 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's 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 centered at the origin with radius 1 is called the unit circle.
- Thought of as a great circle of the 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 center of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

Chord properties

Chords are equidistant from the center of a circle if and only if they are equal in length.
The perpendicular bisector of a chord passes through the center of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:
- A perpendicular line from the center of a circle bisects the chord.
- The line segment (circular segment) through the center 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 supplemental.
- For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
An inscribed angle subtended by a diameter is a right angle.
The diameter is the longest chord of the circle.

Sagitta properties

The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the circumference 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 which will fit around the two lines:

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

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 it is part of a diameter of the circle. Since the diameter is twice the radius, the “missing” part of the diameter is (2r − 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 (2r − x)x = (y/2)². Solving for r, we find the required result.

Tangent properties

The line drawn perpendicular to a radius through the end point of the radius is a tangent to the circle.
A line drawn perpendicular to a tangent through the point of contact with a circle passes through the center 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.

Theorems

Secant-secant theorem

Inscribed angles

Apollonius circle

Apollonius' definition of a circle: d₁/d₂ = constant.

Apollonius of Perga showed that a circle may also be defined as the set of points in plane having a constant ratio of distances to two fixed foci, A and B. That circle is sometimes said to be drawn about two points^[3].

The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:

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

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to $180^{\circ}$ , the angle CPD is exactly $90^{\circ}$ , i.e., a right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.

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 Apollonius circle for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one:

$|[A,B;C,P]| = 1.\$

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

Generalized circles

Notes

^ Chronology for 30000 BC to 500 BC
^ Squaring the circle
^ Harkness, James (1898). Introduction to the theory of analytic functions. London, New York: Macmillan and Co.. pp. 30. http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01680002.

References

^ Pedoe, Dan (1988). Geometry: a comprehensive course. Dover.
"Circle" in The MacTutor History of Mathematics archive

External links

Wikimedia Commons has media related to: Circle geometry

Circle formulas at Geometry Atlas.
Interactive Java applets for the properties of and elementary constructions involving circles.
Interactive Standard Form Equation of Circle Click and drag points to see standard form equation in action
Munching on Circles at cut-the-knot
Ron Blond homepage - interactive applets
calculate circumference and area with your own values
MathAce » Circles MathAce's article about circles - has a good in-depth explanation of unit circles and transforming circular equations.

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

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Translations: Circle

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Dansk (Danish)
n. - cirkel, rundkreds, omkreds, kreds, rotunde
v. tr. - kredse om, tegne cirkel om, omgive
v. intr. - kredse

Nederlands (Dutch)
kring, cirkel(-), rotonde, balkon (theater), slagcirkel, cyclus, omcirkelen, omlijnen

Français (French)
n. - cercle, rond, groupe, (Théât) balcon
v. tr. - tourner autour de, graviter autour de, faire le tour de, tourner autour de (qn, animal), encercler
v. intr. - tourner en rond autour de, décrire des cercles au-dessus de

Deutsch (German)
n. - Kreis, Rang, Kreisel
v. - einkreisen, umkreisen

Ελληνική (Greek)
n. - κύκλος, ανακύκληση, κοινωνικός κύκλος, κοινωνικός περίγυρος, εξώστης θεάτρου
v. - ζώνω, κυκλώνω, περικυκλώνω, περιτριγυρίζω, περιστρέφομαι, στριφογυρίζω

idioms:

high circles υψηλοί κύκλοι, της καλής κοινωνίας

Italiano (Italian)
circondare, cerchia, circolo, cerchio, balconata, galleria, rotonda, circolare

idioms:

come full circle/ turn full circle ritornare al punto di partenza
go round in a circle girare in tondo

Português (Portuguese)
n. - círculo (m), circunferência (f), anel (m), coroa (f), giro (m), ciclo (m), rodeio (m), órbita (f), âmbito (m), ciclo (m) social, divisão (f) territorial
v. - circundar, girar em torno, mover-se em círculos

idioms:

come/turn full circle voltar ao início
dress circle balcão (m) nobre
go round in a circle não progredir
high circles altas rodas (f pl)
inner circle grupo (m) fechado

Русский (Russian)
описывать круги, обводить кружком, круг, цикл, бельэтаж

idioms:

come/turn full circle описать полный круг
dress circle бельэтаж
go round in a circle вертеться как белка в колесе
high circles высшие круги
inner circle наиболее близкие друзья, приближенные

Español (Spanish)
n. - círculo, corro, rueda, anillo, piso principal del teatro, rotonda, circular
v. tr. - cercar, rodear
v. intr. - poner un círculo alrededor de, rodearse

Svenska (Swedish)
n. - cirkel, omkrets, kretsgång, full serie, krets, rad (teat.), rund stensättning (arkeol.)
v. - omge, gå/fara omkring, ringa in, kretsa

中文（简体）(Chinese (Simplified))
圆周, 循环, 社交圈, 包围, 环绕, 盘旋, 旋转, 流传, 环行

中文（繁體）(Chinese (Traditional))
n. - 圓周, 循環, 社交圈
v. tr. - 包圍, 環繞
v. intr. - 盤旋, 旋轉, 流傳, 環行

한국어 (Korean)
n. - 원, 주기, 집단
v. tr. - 선회하다, 에워싸다, 우회하다
v. intr. - 돌다

日本語 (Japanese)
n. - 円, 仲間, 範囲, 周期, 桟敷, 圏, 輪, 集団, 全系統, 悪循環
v. - 回る, 一周する, 丸で囲む, 回転する

العربيه (Arabic)
‏(الاسم) دائرة, حلقه (فعل) حام, طاف, أحاط, دور‏

עברית (Hebrew)
n. - ‮מחזור, חוג, טבעת, מעגל, עיגול, גוש מושבים בתיאטרון‬
v. tr. - ‮הקיף‬
v. intr. - ‮הסתובב, חג‬

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Best of the Web: circle

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Some good "circle" pages on the web:

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commtechlab.msu.edu

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mathworld.wolfram.com

Shopping: circle

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circle

Contents

Further terminology

History

Analytic results

Length of circumference

Area enclosed

Equation

Tangent lines

Properties

Chord properties

Sagitta properties

Tangent properties

Theorems

Inscribed angles

Apollonius circle

Cross-ratios

Generalized circles

See also

Notes

References

External links