Circle

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Circle

A circle is a simple shape

Shape

The shape of an object located in some space is the part of that space occupied by the object, as determined by its external boundary ? abstracting from other properties such as colour, content, and material composition, as well as from the object's other spatial properties ....
of Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
consisting of those points

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
in a plane

Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
which are the same distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
from a given point called the center. The common distance of the points of a circle from its center is called its radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
. A diameter
Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle....
is a line segment

Line segment

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Further terminology

A chord
Chord (geometry)

A chord of a curve is a geometry line segment whose endpoints both lie on the curve.A secant or a secant line is the line extension of a chord....
of a circle is a line segment whose two endpoints lie on the circle. The diameter, passing through the circle's centre, is the largest chord in a circle. A tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
to a circle is a straight line that touches the circle at a single point. A secant
Secant line

A secant line of a curve is a line that intersects two Point s on the curve. The word secant comes from the Latin secare, for to cut....
is an extended chord: a straight line cutting the circle at two points.

An arc
Arc (geometry)

In geometry, an arc is a closed set segment of a differentiable curve in the two-dimensional manifold; for example, a circular arc is a segment of the circumference of a circle....
of a circle is any connected

Connected space

In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open subsets....
part of the circle's perimeter. A sector
Circular sector

A circular sector or circle sector, is the portion of a circle enclosed by two radius and an Arc , where the smaller area is known as the minor sector and the larger being the major sector....
is a region bounded by two radii and an arc lying between the radii, and a segment
Circular segment

In geometry, a circular segment is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant line or a chord ....
is a region bounded by a chord and an arc lying between the chord's endpoints.

History

The circle has been known since before the beginning of recorded history. It is the basis for the wheel

Wheel

A wheel is a circular device that is capable of rotating on its axis, facilitating movement or transportation whilst supporting a load , or performing labour in machines....
which, with related inventions such as gear

Gear

A gear is a component within a Transmission device that transmits rotational force to another gear or device. A gear is different from a pulley in that a gear is a round wheel that has linkages that mesh with other gear teeth, allowing force to be fully transferred without slippage....
s, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus.

Early science

Science

In its broadest sense, science refers to any systematic knowledge or practice. In its more usual restricted sense, science refers to a system of acquiring knowledge based on scientific method, as well as to the organized body of knowledge gained through such research....
, particularly geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
and Astrology and astronomy

Astrology and astronomy

Astrology and astronomy are historically one and the same discipline , and were only gradually recognized as separate in Western World 17th century philosophy ....
, was connected to the divine for most medieval scholars

History of science in the Middle Ages

In the Middle Ages, science progressed dramatically from the time of Ancient history in areas as diverse as astronomy, medicine, and mathematics. Whereas the ancient cultures of the world had developed many of the foundations of science, it was during the Middle Ages that the scientific method was born and science became a formal discipline separa...
, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.

Some highlights in the history of the circle are:

1700 BC – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of .
300 BC – Book 3 of Euclid's Elements
Euclid's Elements

Euclid's Elements is a mathematics and geometry treatise consisting of 13 books written by the Greek mathematics Euclid in Alexandria circa 300 BC....
deals with the properties of circles.
1880 – Lindemann
Ferdinand von Lindemann

Carl Louis Ferdinand von Lindemann was a Germany mathematician, noted for his proof, published in 1882, that pi is a transcendental number, i.e., it is not a zero of any polynomial with rational number coefficients....
proves that is transcendental, effectively settling the millennia-old problem of squaring the circle
Squaring the circle

Squaring the circle is a problem proposed by classical antiquity geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge....
.

Analytic results

Length of circumference

The ratio of a circle's circumference

Circumference

The circumference is the distance around a closed curve. Circumference is a kind of perimeter....
to its diameter

Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle....
is p

Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
(pi), a constant

Mathematical constant

A mathematical constant is a number, usually a real number, that arises naturally in mathematics. Unlike physical constants, mathematical constants are defined independently of physical measurement....
that takes the same value (approximately 3.1416) for all circles. Thus the length of the circumference (c) is related to the radius (r) by

or equivalently to the diameter (d) by

Area enclosed

The area enclosed by a circle is multiplied by the radius squared:

Equivalently, denoting diameter by d,

that is, approximately 79% of the circumscribing square (whose side is of length d).

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations

Calculus of variations

Calculus of variations is a field of mathematics that deals with functional , as opposed to ordinary calculus which deals with function . Such functionals can for example be formed as integrals involving an unknown function and its derivatives....
, namely the isoperimetric inequality.

Equation

In an x-y Cartesian coordinate system

Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
, the circle with center (a, b) and radius r is the set of all points (x, y) such that

This equation
Equation

An equation is a mathematics Proposition, in table of mathematical symbols, that two things are exactly the same . Equations are written with an equal sign, as in...
of the circle follows from the Pythagorean theorem

Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centered at the origin (0, 0), then the equation simplifies to

The equation can be written in parametric form using the trigonometric function

Trigonometric function

In mathematics, the trigonometric functions are function s of an angle. They are important in the trigonometry of Triangle and modeling Periodic function, among many other applications....
s sine and cosine as

where t is a parametric variable, interpreted geometrically as the angle that the ray from the origin to (x, y) makes with the x-axis. Alternatively, in stereographic coordinates

Stereographic projection

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane . The projection is defined on the entire sphere, except at one point — the projection point....
, the circle has a parametrization

In homogeneous coordinates

Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand M?bius in his 1827 work Der barycentrische Calc?l, allow affine transformations to be easily represented by a matrix....
each conic section

Conic section

File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane ....
with equation of a circle is of the form

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

Circular points at infinity

In projective geometry, the circular points at infinity in the complex projective plane are and .Here the coordinates are homogeneous coordinates ; so that the line at infinity is defined by z = 0....
.

In polar coordinates the equation of a circle is

where a is the radius of the circle, r₀ is the distance from the origin to the centre of the circle, and f is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle. For a circle centred at the origin, this reduces to simply r = a.

In the complex plane

Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
, a circle with a center at c and radius (r) has the equation . Since , the slightly generalized equation for real p, q and complex g is sometimes called a generalised circle

Generalised circle

A circle G is the Set of Point p in a plane that lie at radius r from a center point ?.Using the complex plane, we can treat ? as a complex number and circle G as a set of complex numbers....
. 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

Hence, the slope

Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
of a circle at (x₁, y₁) is given by:

More generally, the slope

Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
at a point (x, y) on the circle , i.e., the circle centered at (a, b) with radius r units, is given by

provided that .

Properties

The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)
The circle is a highly symmetric shape: every line through the center forms a line of reflection symmetry
Reflection symmetry

The triangles with this symmetry are isosceles. The quadrilaterals with this symmetry are the kite s and the isosceles trapezoids.For each line or plane of reflection, the symmetry group is isomorphic with Cs , one of the three types of order two , hence algebraically C2....
and it has rotational symmetry
Rotational symmetry

File:The armoured triskelion on the flag of the Isle of Man.svgGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation....
around the center for every angle. Its symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
is the orthogonal group
Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
O(2,R). The group of rotations alone is the circle group
Circle group

In mathematics, the circle group, denoted by T , is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane....
T.
All circles are similar.

A circle's circumference and radius are proportional
Proportionality (mathematics)

In mathematics, two quantity are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio....
.
The area enclosed and the square of its radius are proportional
Proportionality (mathematics)

In mathematics, two quantity are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio....
.

The constants
Mathematical constant

A mathematical constant is a number, usually a real number, that arises naturally in mathematics. Unlike physical constants, mathematical constants are defined independently of physical measurement....
of proportionality are 2p
Pi

Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
and p, respectively.

The circle centered at the origin with radius 1 is called the unit circle
Unit circle

In mathematics, a unit circle is a circle with a 1 radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane....
.

Thought of as a great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
of the unit sphere, it becomes the Riemannian circle
Riemannian circle

In metric space theory and Riemannian geometry, the term Riemannian circle refers to a great circle equipped with its great-circle distance. In more detail, the term refers to the circle equipped with its intrinsic Riemannian metric of a compact 1-dimensional manifold of total length 2π, as opposed to the extrinsic metric obtaine...
.

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 are equidistant from the center of a circle if and only if they are equal in length.
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
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
(circular segment
Circular segment

In geometry, a circular segment is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant line or a chord ....
) through the center bisecting a chord is perpendicular to the chord.

If a central angle and an inscribed angle
Inscribed angle

In geometry, an inscribed angle is formed when two secant lines of a circle intersect on the circle.Typically, it is easiest to think of an inscribed angle as being defined by two Chord of the circle sharing an endpoint....
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 the longest chord of the circle.

Sagitta properties

The sagitta (also known as the versine
Versine

The versed sine, also called the versine and, in Latin, the sinus versus or the sagitta , is a trigonometric function versin .Although the versine function appeared in some of the earliest trigonometric tables and was once widespread , it is now little-used....
) 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
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
can be used to calculate the radius of the unique circle which will fit around the two lines:

Another proof of this result which relies only on two 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 (2r - 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 (2r - x)x = (y/2)². Solving for r, we find the required result.

Tangent properties

The line drawn perpendicular to a radius through the end point of the radius is a tangent to the circle.
A line drawn perpendicular to a tangent through the point of contact with a circle passes through the center of the circle.
Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.

Theorems

The chord theorem states that if two chords, CD and EB, intersect at A, then CA×DA = EA×BA.
If a tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
from an external point D meets the circle at C and a secant
Secant line

A secant line of a curve is a line that intersects two Point s on the curve. The word secant comes from the Latin secare, for to cut....
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
Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
then l = v2 × 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

Inscribed angle

In geometry, an inscribed angle is formed when two secant lines of a circle intersect on the circle.Typically, it is easiest to think of an inscribed angle as being defined by two Chord of the circle sharing an endpoint....
(examples are the blue and green angles in the figure) is exactly half the corresponding central angle

Central angle

A central angle is an angle whose vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an Arc between those two points whose angle is equal to the central angle itself....
(red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter

Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle....
is a right angle

Right angle

In geometry and trigonometry, a right angle is an angle of 90 degree s, corresponding to a quarter turn . It can be defined; as the angle such that twice that angle amounts to a half turn, or 180?....
(since the central angle is 180 degrees).

Apollonius circle

Apollonius of Perga

Apollonius of Perga [Pergaeus] was a Greeks geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and Ren? Descartes....
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:

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to , the angle CPD is exactly , i.e., a right angle

Right angle

Cross-ratios

A closely related property of circles involves the geometry of the cross-ratio

Cross-ratio

In mathematics, the cross-ratio of a set of four distinct points on the complex plane is given byThis definition can be extended to the entire Riemann sphere by continuous function....
of points in the complex plane

Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
. 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: 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

Unit circle

In mathematics, a unit circle is a circle with a 1 radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane....
in the complex plane.

Generalized circles

If C is the midpoint

Midpoint

The midpoint is the middle Point of a line segment. It is Distance from both endpoints. The formula for determining the midpoint of a segment in the plane, with endpoints and is...
of the segment AB, then the collection of points P satisfying the Apollonius condition

(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. In this sense a line is generalized circle of infinite radius.

Circle

Further terminology

History

Analytic results

Length of circumference

Area enclosed

Equation

Tangent lines

Properties

Chord properties

Sagitta properties

Tangent properties

Theorems

Inscribed angles

Apollonius circle

Cross-ratios

Generalized circles

See also

External links