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"
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: R2, 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
circulusPronunciation
 Rhymes: ɜː(r)kəl
Noun
 : A twodimensional
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 (x1)^2 + y^2 = r^2 is a circle of radius r around the point (1, 0).
 A twodimensional 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.
 Any thin threedimensional
equivalent of the geometric figures.
 Put on your duncecap and sit down on that circle.
 A curve that more or less forms part or all of a circle.
 move in a circle
 Orbit.
 A specific group of
persons.
 inner circle
 circle of friends
 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 oneday match.
Synonyms
 (twodimensional outline geometric figure): coil (not in mathematical use), ring (not in mathematical use), loop (not in mathematical use)
 (twodimensional solid geometric figure): disc/disk (in mathematical and general use), round (not in mathematical use; UK & Commonwealth only)
 (curve): arc, curve
 (orbit): orbit
Translations
twodimensional outline geometric figure
 trreq Albanian
 Arabic: (dā’ira)
 Armenian: շրջան
 trreq Basque
 Breton: kelc'h , kelc'hioù p
 Bulgarian: кръг (krâg) , окръжност (okrâžnost)
 Catalan: cercle
 Chinese: 圓形, 圆形 (yuán xíng)
 Croatian: kružnica
 Czech: kružnice
 Danish: cirkel
 Dutch: cirkel
 Esperanto: cirklo
 Estonian: ring
 Finnish: ympyrä
 French: cercle
 Georgian: წრე (ts‘re)
 German: Kreis
 Greek: κύκλος (kíklos)
 Hebrew: מעגל (ma'agál)
 trreq Hindi
 Hungarian: kör
 Icelandic: hringur
 Indonesian: lingkaran
 Italian: cerchio
 Japanese: 円 (えん, en), 丸 (まる, maru)
 Korean: 원 (wōn)
 Kurdish:
 Latin: circulus
 Latvian: aplis
 Lithuanian: apskritimas
 Malayalam: വട്ടം (vattam), വൃത്തം (vrutham)
 Maltese: ċirku
 trreq Maori
 Mongolian: дугуй (duguy)
 Norwegian: sirkel
 Persian:
 Portuguese: círculo
 Quechua: diyosuun
 Romanian: cerc
 Russian: окружность (okrúžnost’) , круг (krug) (окружность preferred)
 trreq Sanskrit
 Scottish Gaelic: cruinne mf, cuairt , buail , ràth
 Slovak: kružnica
 Slovene: krožnica
 Spanish: círculo
 Swedish: cirkel
 trreq Tamil
 Telugu: వృత్తము
 Thai: (wăen)
 Turkish: daire
 trreq Urdu
 trreq Vietnamese
 Welsh: cylch
disc, twodimensional solid geometric figure
 Arabic: (dā’ira)
 Bulgarian: кръг (krâg) , окръжност (okrâžnost)
 Catalan: disc
 Chinese: 圓, 圆 (yuán)
 Czech: kruh
 Danish: cirkel
 Dutch: cirkel
 French: disque
 Georgian: წრე (ts‘re)
 Greek: κύκλος (kíklos)
 Hebrew: מעגל (ma'agál)
 Hungarian: körlap
 Italian: disco
 Japanese: 円形のもの (えんけいのもの,enkei no mono)
 Latin: circulus
 Portuguese: círculo
 Russian: круг (krug) , окружность (okrúžnost’) (круг preferred)
 Scottish Gaelic: cruinne mf, cuairt , buail , ràth
 Slovak: kruh
 Slovene: krog
 Spanish: círculo
 Swedish: cirkel , cirkelskiva
 Thai: (duang)
threedimensional geometric figure
curve
orbit
 Breton: kelc'h , kelc'hioù p
 Bulgarian: орбита (orbita)
 Catalan: òrbita
 Chinese: 軌道, 轨道 (guǐ dào)
 Danish: kredsløb
 Dutch: baan
 Finnish: rata (2)
 Georgian: ორბიტი (orbiti)
 Hebrew: מסלול (maslul)
 Italian: orbita
 Latin: orbis
 Malayalam: ഭ്രമണ പഥം (bhramaNa patham)
 Portuguese: círculo
 Russian: орбита (orbíta)
 Slovene: krožnica
 Spanish: órbita
 Telugu: కక్ష్య
group of persons
 Arabic: (ħálqa)
 Chinese: 圈子, 圈子 (quān zi)
 Czech: kruh
 Danish: kreds
 Dutch: kring, groep
 Esperanto: rondo
 Finnish: piiri
 Georgian: წრე (ts‘re)
 German: Zirkel
 Greek: κύκλος (kíklos)
 Hebrew: חוג (khug)
 Italian: circolo , gruppo
 Japanese: サークル (sākuru)
 Latin: corona
 Malayalam: വൃത്തം (vrutham)
 Portuguese: círculo
 Russian: круг (krug)
 Scottish Gaelic: còmhlan
 Slovene: krog
 Spanish: círculo , grupo
 Swedish: krets
Verb
Translations
travel around along a curved path
surround
place or mark a circle around
 Chinese: 畫圓圈, 画圆圈 (huà yuán quān)
 Danish: sætte ring om
 Dutch: omcirkelen
 Finnish: ympäröidä, ympyröidä (draw a circle)
 German: einkreisen
 Hebrew: להקיף בעיגול (le'haqyf be'igul)
 Italian: cerchiare
 Portuguese: circular
 Slovak: zakrúžkovať
 Spanish: circular
 Swedish: ringa in, inringa
travel in circles
 ttbc Breton: kelc'hiañ, gronnañ
 ttbc Indonesian: mengelilingi, mengitari
 ttbc Interlingua: circular
 ttbc Irish: ciorcal
 ttbc Polish: krążyć
 ttbc Romanian: încercui
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 xy 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 xaxis. 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 zc^2 = r^2. Since zc^2 =
z\overline\overlinezc\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 (xa)^2 +(yb)^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
The numeric value of \pi never changes. In modern
English, it is (as in apple pie).
Area enclosed
 The area enclosed by a circle is the radius squared, multiplied by \pi.
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.
 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.
 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*rx) 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*rx)(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. (tangentsecant theorem)
 If two secants, DG and DE, also cut the circle at H and F respectively, then DH×DG=DF×DE. (Corollary of the tangentsecant 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 secantsecant 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.
Crossratios
A closely related property of circles involves
the geometry of the crossratio 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 crossratio is equal to one:
 [A,B;C,P] = 1.
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
See also
External links
 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
 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 cuttheknot
 Ron Blond homepage  interactive applets
 calculate circumference and area with your own values
 MathAce » Circles MathAce's article about circles  has a good indepth explanation of unit circles and transforming circular equations.
circle in Contenese: 圓形
circle in Arabic: دائرة
circle in Asturian: Círculu
circle in Aymara: Muyu
circle in Min Nan: Îⁿhêng
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, wegroup, wheel, wheel around, whirl, wind, wreath, wreathe, wreathe around,
wristband, wristlet, zodiac, zone