Circle
A circle is the set of points in a plane that are equidistant from a given point 
. The distance 
from the center is called the radius, and the point is called the center. Twice the radius is known as the diameter 
. The angle a circle subtends from its center is a full angle, equal to 
or 
radians.
A circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area.
The perimeter 
of a circle is called the circumference, and is given by
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(1)
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This can be computed using calculus using the formula for arc length in polar coordinates,
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(2)
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but since , this becomes simply
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(3)
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The circumference-to-diameter ratio 
for a circle is constant as the size of the circle is changed (as it must be since scaling a plane figure by a factor 
increases its perimeter by 
), and 
also scales by 
. This ratio is denoted 
(pi), and has been proved transcendental.
Knowing 
, the area of the circle can be computed either geometrically or using calculus. As the number of concentric strips increases to infinity as illustrated above, they form a triangle, so
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(4)
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This derivation was first recorded by Archimedes in Measurement of a Circle (ca. 225 BC).
If the circle is instead cut into wedges, as the number of wedges increases to infinity, a rectangle results, so
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(5)
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From calculus, the area follows immediately from the formula
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(6)
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again using polar coordinates.
A circle can also be viewed as the limiting case of a regular polygon with inradius 
and circumradius as the number of sides 
approaches infinity (a figure technically known as an apeirogon). This then gives the circumference as
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(7)
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|
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(8)
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and the area as
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(9)
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(10)
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which are equivalently since the radii 
and 
converge to the same radius as 
.
Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "
-sphere," with geometers referring to the number of coordinates in the underlying space and topologists referring to the dimension of the surface itself (Coxeter 1973, p. 125). As a result, geometers call the circumference of the usual circle the 2-sphere, while topologists refer to it as the 1-sphere and denote it 
.
The circle is a conic section obtained by the intersection of a cone with a plane perpendicular to the cone's symmetry axis. It is also a Lissajous curve. A circle is the degenerate case of an ellipse with equal semimajor and semiminor axes (i.e., with eccentricity 0). The interior of a circle is called a disk. The generalization of a circle to three dimensions is called a sphere, and to 
dimensions for 
a hypersphere.
The region of intersection of two circles is called a lens. The region of intersection of three symmetrically placed circles (as in a Venn diagram), in the special case of the center of each being located at the intersection of the other two, is called a Reuleaux triangle.
In Cartesian coordinates, the equation of a circle of radius 
centered on 
is
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(11)
|
In pedal coordinates with the pedal point at the center, the equation is
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(12)
|
The circle having 
as a diameter is given by
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(13)
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The parametric equations for a circle of radius 
can be given by
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(14)
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(15)
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The circle can also be parameterized by the rational functions
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(16)
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(17)
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but an elliptic curve cannot.
The plots above show a sequence of normal and tangent vectors for the circle.
The arc length 
, curvature 
, and tangential angle 
of the circle with radius 
represented parametrically by (◇) and (◇) are
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(18)
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(19)
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(20)
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The Cesàro equation is
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(21)
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In polar coordinates, the equation of the circle has a particularly simple form.
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(22)
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is a circle of radius 
centered at origin,
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(23)
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is circle of radius 
centered at , and
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(24)
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is a circle of radius 
centered on 
.
The equation of a circle passing through the three points 
for 
, 2, 3 (the circumcircle of the triangle determined by the points) is
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(25)
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The center and radius of this circle can be identified by assigning coefficients of a quadratic curve
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(26)
|
where 
and 
(since there is no 
cross term). Completing the square gives
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(27)
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The center can then be identified as
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(28)
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(29)
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and the radius as
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(30)
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where
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(31)
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(32)
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(33)
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(34)
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Four or more points which lie on a circle are said to be concyclic. Three points are trivially concyclic since three noncollinear points determine a circle.
In trilinear coordinates, every circle has an equation of the form
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(35)
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with 
(Kimberling 1998, p. 219).
The center 
of a circle given by equation (35) is given by
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(36)
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(37)
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|
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(38)
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(Kimberling 1998, p. 222).
In exact trilinear coordinates 
, the equation of the circle passing through three noncollinear points with exact trilinear coordinates 
, , and 
is
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(39)
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(Kimberling 1998, p. 222).
An equation for the trilinear circle of radius 
with center 
is given by Kimberling (1998, p. 223).
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