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