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
|
(1)
|
This can be computed using
calculus
using the formula for
arc length
in
polar
coordinates
,
|
(2)
|
but since
, this becomes simply
|
(3)
|
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
|
(4)
|
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
|
(5)
|
From
calculus
, the area follows immediately from the
formula
|
(6)
|
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
and the
area
as
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
|
(11)
|
In
pedal coordinates
with the
pedal
point
at the center, the equation is
|
(12)
|
The circle having
as a diameter is given by
|
(13)
|
The
parametric equations
for a circle of
radius
can be given by
The circle can also be parameterized by the rational functions
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
The
Cesàro equation
is
|
(21)
|
In
polar coordinates
, the equation of the circle
has a particularly simple form.
|
(22)
|
is a circle of
radius
centered at
origin
,
|
(23)
|
is circle of
radius
centered at
, and
|
(24)
|
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
|
(25)
|
The
center
and
radius
of this circle can be identified by assigning coefficients of a
quadratic
curve
|
(26)
|
where
and
(since there is no
cross term).
Completing
the square
gives
|
(27)
|
The
center
can then be identified as
and the
radius
as
|
(30)
|
where
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
|
(35)
|
with
(Kimberling 1998, p. 219).
The center
of a circle given by equation (
35
) is given by
(Kimberling 1998, p. 222).
In
exact trilinear coordinates
, the equation of the
circle passing through three noncollinear points with
exact
trilinear coordinates
,
, and
is
|
(39)
|
(Kimberling 1998, p. 222).
An equation for the trilinear circle of radius
with center
is given by Kimberling (1998, p. 223).