Dictionary:

circle

  (sûr'kəl)

(Click to enlarge)

circle

(Academy Artworks)

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

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

To move in a circle. See synonyms at turn.

circle the wagons

1. To take a defensive position; become defensive.

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

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.

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.

noun

1. 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.
2. A course, process, or journey that ends where it began or repeats itself: circuit, cycle, orbit, round, tour, turn. See repetition.
3. A group of people sharing an interest, activity, or achievement: crowd, group, set. See group.
4. A particular social group: clique, coterie, crowd, set. Informal bunch, gang. See group.
5. 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

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

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.

For more information on circle, visit Britannica.com.

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.

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.

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.

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).

Circle illustration

This article is about the shape and mathematical concept of circle. For other uses, see Circle (disambiguation).

In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

Circles are simple closed curves which divide the plane into an interior and exterior. The circumference of a circle means the length of the circle, and the interior of the circle is called a disk. An arc is any continuous portion of a circle.

A circle is a special ellipse in which the two foci coincide (i.e., are the same point). Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

Analytic results

Equation of a circle

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

If the circle is centred at the origin (0, 0), then this formula can be simplified to

and its tangent will be

where x₁, y₁ are the coordinates of the common point.

When expressed in parametric equations, (x, y) can be written using the trigonometric functions sine and cosine as

where t is a parametric variable, understood as the angle the ray to (x, y) makes with the x-axis.

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

ax² + ay² + 2b₁xz + 2b₂yz + cz² = 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

In the complex plane, a circle with a centre at c and radius r has the equation | z - c | ² = r². Since , the slightly generalized equation for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles.

Slope

The slope of a circle at a point (x, y) can be expressed with the following formula, assuming the centre is at the origin and (x, y) is on the circle:

More generally, the slope at a point (x, y) on the circle (x - a)² + (y - b)² = r², i.e., the circle centred at (a, b) with radius r units, is given by

provided that y≠b, of course.

Pi (π)

Pi or π is the ratio of a circle's Circumference to its Diameter.

The numeric value of π never changes.

π is always approximately 3.14159.

In modern English, it is pronounced /paɪ/ (as in apple pie).

Circumference

Main article: Circumference

• Length of a circle's circumference is

• Alternate formula for circumference:

Given that the ratio circumference c to the Area A is

The r and the π can be canceled, leaving

Therefore solving for c:

So the circumference is equal to 2 times the area, divided by the radius. This can be used to calculate the circumference when a value for π cannot be computed.

Diameter

Main article: Diameter

The diameter of a circle is a straight line through the center of the circle touching the circle at both sides.

The diameter of a circle is double its radius.

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 the radius squared, multiplied by π.

Using a square with side lengths equal to the diameter of the circle, then dividing the square into four squares with side lengths equal to the radius of the circle, take the area of the smaller square and multiply by π.

that is, approximately 79% of the circumscribing square.

Properties

• The circle is the shape with the highest area for a given length of perimeter. (See Isoperimetry)
• 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'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 centred at the origin with radius 1 is called the unit circle.

Chord properties

• Chords equidistant from the centre of a circle are equal (length).
• Equal (length) chords are equidistant from the centre.
• The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:
  • A perpendicular line from the centre of a circle bisects the chord.
  • The line segment (Circular 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 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 longest chord of the circle.

Sagitta properties

• The sagitta is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the circumference of the circle.
• Given the length of a chord, y, 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 2 lines :

written by priyam saini

Tangent properties

• The line drawn perpendicular to the end point of a radius is a tangent to the circle.
• A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
• Tangents drawn from a point outside the circle are equal in length.
• Two tangents can always be drawn from a point outside of the circle.

Theorems

Secant-secant theorem

See also: Power of a point

• The chord theorem states that if two chords, CD and EF, intersect at G, then CG×DG = EG×FG. (Chord theorem)
• If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then DC² = DG×DE. (tangent-secant theorem)
• If two secants, DG and DE, also cut the circle at H and F respectively, then DH×DG = DF×DE. (Corollary of the tangent-secant theorem)
• The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent chord property)
• If the angle subtended by the chord at the centre is 90 degrees then l = √(2) × r, where l is the length of the chord and r is the radius of the circle.
• If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.

Inscribed angles

Inscribed angle theorem

An inscribed angle ψ is exactly half of the corresponding central angle θ (see Figure). Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles ψ in the Figure). Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle.

An alternative definition of a circle

Apollonius' definition of a circle

Apollonius of Perga showed that a circle may also be defined as the set of points having a constant ratio of distances to two foci, A and B.

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

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to , the angle CPD is exactly , 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.
As a point of clarification, note that C and D are determined by A, B, and the desired ratio; i.e. A and B are not arbitrary points lying on an extension of the diameter of an existing circle.

Calculating the parameters of a circle

Early science, particularly geometry and astronomy/astrology, was connected to the divine for most medieval scholars. The compass in this 13th century manuscript is a symbol of God's act of Creation, as many believed that there was something intrinsically "divine" or "perfect" that could be found in circles

The Twelve-Mile Circle is an arc of a circle with a twelve-mile radius, with the center of the circle in the center of the town of New Castle, Delaware.

Given three non-collinear points lying on the circle

Radius

The radius of the circle is given by

Center

The center of the circle is given by

where

Plane unit normal

A unit normal of the plane containing the circle is given by

Parametric Equation

Given the radius, r , center, P_c, a point on the circle, P₀ and a unit normal of the plane containing the circle, , the parametric equation of the circle starting from the point P₀ and proceeding counterclockwise is given by the following equation:

References

Notes

See also

Wikimedia Commons has media related to:

Circle

• Area enclosed by a circle
• Ball
• Circular sector
• Degree (angle)
• Descartes' theorem
• Directional statistics
• Ellipse
• Isoperimetric theorem
• List of circle topics
• Pi
• Sphere
• Squircle
• Unit circle
• von Mises distribution
• Solar symbol
• Circumscribed circle

External links

• 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
• Clifford's Circle Chain Theorems. Step by step presentation of the first theorem. Clifford discovered, in the ordinary Euclidean plane, a "sequence or chain of theorems" of increasing complexity, each building on the last in a natural progression by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
• Munching on Circles at cut-the-knot
• Ron Blond homepage - interactive applets
• calculate circumference and area with your own values

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

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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