AskDefine | Define circle

Dictionary Definition

circle

Noun

1 ellipse in which the two axes are of equal length; a plane curve generated by one point moving at a constant distance from a fixed point; "he calculated the circumference of the circle"
2 an unofficial association of people or groups; "the smart set goes there"; "they were an angry lot" [syn: set, band, lot]
3 something approximating the shape of a circle; "the chairs were arranged in a circle"
4 movement once around a course; "he drove an extra lap just for insurance" [syn: lap, circuit]
5 a road junction at which traffic streams circularly around a central island; "the accident blocked all traffic at the rotary" [syn: traffic circle, rotary, roundabout]
6 street names for flunitrazepan [syn: R-2, Mexican valium, rophy, rope, roofy, roach, forget me drug]
7 a curved section or tier of seats in a hall or theater or opera house; usually the first tier above the orchestra; "they had excellent seats in the dress circle" [syn: dress circle]
8 any circular or rotating mechanism; "the machine punched out metal circles" [syn: round]

Verb

1 travel around something; "circle the globe"
2 move in circles [syn: circulate]
3 be around; "Developments surround the town"; "The river encircles the village" [syn: surround, environ, encircle, round, ring]
4 form a circle around; "encircle the errors" [syn: encircle]

User Contributed Dictionary

English

Etymology

circulus

Pronunciation

Noun

  1. : A two-dimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from another point.
    The set of all points (x, y) such that (x-1)^2 + y^2 = r^2 is a circle of radius r around the point (1, 0).
  2. A two-dimensional geometric figure, a disk, consisting of the set of all those points of a plane at a distance less than or equal to a fixed distance from another point.
  3. Any thin three-dimensional equivalent of the geometric figures.
    Put on your dunce-cap and sit down on that circle.
  4. A curve that more or less forms part or all of a circle.
    move in a circle
  5. Orbit.
  6. A specific group of persons.
    inner circle
    circle of friends
  7. A line comprising two semicircles of 30 yds radius centred on the wickets joined by straight lines parallel to the pitch used to enforce field restrictions in a one-day match.

Synonyms

  • (two-dimensional outline geometric figure): coil (not in mathematical use), ring (not in mathematical use), loop (not in mathematical use)
  • (two-dimensional solid geometric figure): disc/disk (in mathematical and general use), round (not in mathematical use; UK & Commonwealth only)
  • (curve): arc, curve
  • (orbit): orbit

Translations

two-dimensional outline geometric figure
disc, two-dimensional solid geometric figure
three-dimensional geometric figure
curve
orbit
group of persons

Verb

  1. To travel around along a curved path.
  2. To surround.
  3. To place or mark a circle around.
    Circle the jobs that you are interested in applying for.
  4. To travel in circles.
    Vultures circled overhead.

Translations

travel around along a curved path
surround
place or mark a circle around
travel in circles

Extensive Definition

Circles are simple shapes of Euclidean geometry. A circle consists of those points in a plane which are at a constant distance, called the radius, from a fixed point, called the center. A circle with center A is sometimes denoted by the symbol .
A chord of a circle is a line segment whose both endpoints lie on the circle. A diameter is a chord passing through the center. The length of a diameter is twice the radius. A diameter is the largest chord in a circle.
Circles are simple closed curves which divide the plane into an interior and an exterior. The circumference of a circle is the perimeter of the circle, and the interior of the circle is called a disk. An arc is any connected part of a circle.
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.

Analytic results

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.
The equation of the circle follows from the Pythagorean theorem applied to any point on the circle. If the circle is centred at the origin (0, 0), then this formula can be simplified to
x^2 + y^2 = r^2. \!\
When expressed in parametric equations, (x, y) can be written using the trigonometric functions sine and cosine as
x = a+r\,\cos t,\,\!
y = b+r\,\sin t\,\!
where t is a parametric variable, understood as many the angle the ray to (x, y) makes with the x-axis. Alternatively, in stereographic coordinates, the circle has a parametrization
x = a + r \frac
y = b + r \frac
In homogeneous coordinates each conic section with equation of a circle is
\ 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.\,
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-\overlinez-c\overline+c\overline, the slightly generalised equation pz\overline + gz + \overline = 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

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 = - \frac.
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 = \frac,
provided that y \neq b.

Pi ( \pi )

details Pi
Pi or π is the ratio of a circle's circumference to its diameter.
The numeric value of \pi never changes. In modern English, it is (as in apple pie).

Area enclosed

Area = r^2 \cdot \pi
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 \pi. A = \frac \approx 07854 \cdot d^2, that is, approximately 79% of the circumscribing square.
The circle is the plane curve enclosing the maximum area for a given arclength. This relates the circle to a problem in the calculus of variations.

Properties

  • The circle is the shape with the largest 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 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.
  • The circle centered 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 center of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

Chord properties

  • Chords equidistant from the center of a circle are equal (length).
  • Equal (length) chords are equidistant from the center.
  • 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 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 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+ \frac.
Another proof of this result which relies only on 2 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 (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 (2*r-x)(x)=(y/2)^2. Solving for r, we find:
r=\frac+ \frac.
as required.

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

  • The chord theorem states that if two chords, CD and EB, intersect at A, then CA×DA = EA×BA. (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 center 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

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

Apollonius circle

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.
The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:
\frac = \frac.
Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to 180^, the angle CPD is exactly 90^, 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

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition
\frac = \frac   (1)
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 (1) is called a generalized circle. It may either be a true circle or a line.

References

  • note PedoeGeometry: a comprehensive course

External links

circle in Contenese: 圓形
circle in Arabic: دائرة
circle in Asturian: Círculu
circle in Aymara: Muyu
circle in Min Nan: Îⁿ-hêng
circle in Bosnian: Krug
circle in Bulgarian: Окръжност
circle in Catalan: Cercle
circle in Czech: Kružnice
circle in Welsh: Cylch
circle in Danish: Cirkel
circle in German: Kreis (Geometrie)
circle in Estonian: Ringjoon
circle in Modern Greek (1453-): Κύκλος
circle in Spanish: Círculo
circle in Esperanto: Cirklo
circle in Basque: Zirkulu
circle in Persian: دایره
circle in French: Cercle
circle in Galician: Círculo
circle in Korean: 원 (기하)
circle in Croatian: Kružnica
circle in Indonesian: Lingkaran
circle in Icelandic: Hringur
circle in Italian: Cerchio
circle in Hebrew: מעגל
circle in Haitian: Sèk
circle in Swahili (macrolanguage): Duara
circle in Latin: Circulus
circle in Latvian: Riņķis
circle in Luxembourgish: Krees (Geometrie)
circle in Lithuanian: Apskritimas
circle in Hungarian: Kör
circle in Macedonian: Кружница
circle in Malayalam: വൃത്തം
circle in Malay (macrolanguage): Bulatan
circle in Dutch: Cirkel
circle in Japanese: 円 (数学)
circle in Norwegian: Sirkel
circle in Norwegian Nynorsk: Sirkel
circle in Polish: Okrąg
circle in Portuguese: Círculo
circle in Kölsch: Kriiß (Mattematik)
circle in Quechua: P'allta muyu
circle in Russian: Окружность
circle in Scots: Raing
circle in Simple English: Circle
circle in Slovak: Kružnica
circle in Slovenian: Krožnica
circle in Serbian: Круг
circle in Finnish: Ympyrä
circle in Swedish: Cirkel
circle in Tagalog: Bilog
circle in Tamil: வட்டம்
circle in Thai: รูปวงกลม
circle in Turkish: Çember
circle in Ukrainian: Коло
circle in Yoruba: Ìyípo
circle in Chinese: 圆

Synonyms, Antonyms and Related Words

O, acquaintance, advance, alentours, alternate, ambience, ambit, anklet, annular muscle, annulus, anthelion, antisun, aphelion, apogee, arc, arena, areola, armlet, arsis, ascend, associates, astronomical longitude, aura, aureole, autumnal equinox, back, back up, bailiwick, band, bangle, be here again, beads, beat, begird, belt, belt in, bijou, border, borderland, borderlands, bout, bow, bracelet, breastpin, brooch, budge, bunch, cabal, cadre, camarilla, camp, catacaustic, catenary, caustic, celestial equator, celestial longitude, celestial meridian, cell, chain, change, change place, chaplet, charm, charmed circle, chatelaine, cincture, circuit, circuiteer, circulate, circumambiencies, circumambulate, circumference, circumjacencies, circummigrate, circumnavigate, circumrotate, circumscribe, circumstances, circumvent, circumvolute, circus, clan, class, climb, clique, close the circle, closed circle, colures, come again, come and go, come around, come full circle, come round, come round again, come up again, companions, company, compass, comrades, conchoid, context, cordon, corona, coronet, coterie, countersun, course, crank, crescent, crew, cronies, crook, crowd, crown, curl, curve, cycle, demesne, department, descend, describe a circle, diacaustic, diadem, diastole, dimensions, discus, disk, division, domain, dominion, downbeat, earring, ebb, ecliptic, elite, elite group, ellipse, encincture, encircle, enclose, encompass, engird, ensphere, entourage, environ, environing circumstances, environment, environs, equator, equinoctial, equinoctial circle, equinoctial colure, equinox, eternal return, extension, extent, fairy ring, fellowship, festoon, field, flank, flow, fob, fraternity, friends, full circle, galactic longitude, garland, gem, geocentric longitude, geodetic longitude, gestalt, get over, gird, girdle, girdle the globe, glory, go, go about, go around, go round, go sideways, go the round, great circle, group, gyrate, gyre, habitat, halo, heliocentric longitude, hem, hemisphere, hook, hoop, hyperbola, ingroup, inner circle, intermit, jewel, judicial circuit, junta, junto, jurisdiction, lap, lasso, length, lituus, locket, logical circle, longitude, loop, looplet, lot, lunar corona, lunar halo, magic circle, make a circuit, march, meridian, milieu, mob, mock moon, mock sun, moon dog, mount, move, move over, necklace, neighborhood, nimbus, noose, nose ring, orb, orbit, oscillate, outfit, outposts, outskirts, pale, parabola, paraselene, parhelic circle, parhelion, perigee, perihelion, perimeter, period, periphery, pin, pirouette, pivot, plunge, precinct, precincts, precious stone, progress, province, pulsate, pulse, purlieus, push, radius, rainbow, realm, reappear, recur, regress, reoccur, repeat, retrogress, return, revolution, revolve, rhinestone, ring, ringlet, rise, roll, roll around, rondelle, rotate, rotation, round, round trip, roundel, rounds, run, saucer, scope, screw, series, set, shift, sink, sinus, situation, skirt, small circle, soar, society, solar corona, solar halo, solstitial colure, spell, sphere, sphincter, spin, spiral, stickpin, stir, stone, stream, subside, suburbs, sun dog, surround, surroundings, swing, swivel, systole, thesis, tiara, torque, total environment, tour, tracery, trajectory, travel, turn, turn a pirouette, turn around, turn round, twine around, twist, undulate, upbeat, vernal equinox, vicinage, vicinity, vicious circle, walk, wamble, wampum, wane, we-group, wheel, wheel around, whirl, wind, wreath, wreathe, wreathe around, wristband, wristlet, zodiac, zone
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