# Circle

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This article is about the shape and mathematical concept. For other uses, see Circle (disambiguation).

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**circle**is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point, the centre.

Circles are simple closed curves which divide the plane into two regions, an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.

A circle is a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

## Contents[hide] |

## Terminology

A circle's*diameter*is the length of a line segment whose endpoints lie on the circle and which passes through the centre. This is the largest distance between any two points on the circle. The diameter of a circle is twice the

*radius*, or distance from the centre to the circle's boundary (

*circumference*). The terms 'diameter' and 'radius' also refer to the line segments which fit these descriptions.

A

*chord*is a line segment whose endpoints lie on the circle. A diameter is the longest chord in a circle. A

*tangent*to a circle is a straight line that touches the circle at a single point, while a

*secant*is an extended chord: a straight line cutting the circle at two points.

An

*arc*of a circle is any connected part of the circle's circumference. A

*sector*is a region bounded by two radii and an arc lying between the radii, and a

*segment*is a region bounded by a chord and an arc lying between the chord's endpoints.

## History

The etymology of the word circle is from the Greek, kirkos "a circle," from the base ker- which means to turn or bend. The origin of the word "circus" is closely related as well.The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, and calculus.

Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.

^{[citation needed]}

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 (3.16049...) as an approximate value of π.
^{[1]} - 300 BC – Book 3 of Euclid's Elements deals with the properties of circles.
- In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation.
- 1880 – Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle.
^{[2]}

## Analytic results

### Length of circumference

Further information: Pi

The ratio of a circle's circumference to its diameter is π (pi), an irrational constant (approximately equal to 3.141592654). Thus the length of the circumference *C*is related to the radius

*r*and diameter

*d*by

*C*= 2π

*r*= π

*d*.

### Area enclosed

Main article: Area of a disk

As proved by Archimedes, the area enclosed by a circle is π multiplied by the radius squared:*d*,

*d*). These results can be obtained in several ways, including via integration and by considering the circle as a regular polygon with an infinite number of sides.

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, namely the isoperimetric inequality.

### Equations

#### Cartesian coordinates

In an*x*-

*y*Cartesian coordinate system, the circle with centre coordinates (

*a*,

*b*) and radius

*r*is the set of all points (

*x*,

*y*) such that

*x*−

*a*and

*y*−

*b*. If the circle is centred at the origin (0, 0), then the equation simplifies to

*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, a rational parametrization of the circle is:

*t*to

*r*can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the centre parallel to the

*x*-axis.

In homogeneous coordinates each conic section with equation of a circle is of the form

#### Polar coordinates

In polar coordinates the equation of a circle is:*a*is the radius of the circle,

*r*

_{0}is the distance from the origin to the centre of the circle, and φ 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, i.e.

*r*

_{0}= 0, this reduces to simply

*r*=

*a*. When

*r*

_{0}=

*a*, or when the origin lies on the circle, the equation becomes

*r*= 2*a*cos(θ − φ).

- ,

#### Complex plane

In the complex plane, a circle with a centre at*c*and radius (

*r*) has the equation . In parametric form this can be written

*z*=

*r*

*e*

^{it}+

*c*.

The slightly generalised equation for real

*p*,

*q*and complex

*g*is sometimes called a generalised circle. This becomes the above equation for a circle with , since . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.

### Tangent lines

Main article: Tangent lines to circles

The tangent line through a point *P*on the circle is perpendicular to the diameter passing through

*P*. If

*P*= (

*x*

_{1},

*y*

_{1}) and the circle has centre (

*a*,

*b*) and radius

*r*, then the tangent line is perpendicular to the line from (

*a*,

*b*) to (

*x*

_{1},

*y*

_{1}), so it has the form (

*x*

_{1}−

*a*)x+(

*y*

_{1}−

*b*)y =

*c*. Evaluating at (

*x*

_{1},

*y*

_{1}) determines the value of

*c*and the result is that the equation of the tangent is

- (
*x*_{1}−*a*)*x*+ (*y*_{1}−*b*)*y*= (*x*_{1}−*a*)*x*_{1}+ (*y*_{1}−*b*)*y*_{1}

- (
*x*_{1}−*a*)(*x*−*a*) + (*y*_{1}−*b*)(*y*−*b*) =*r*^{2}.

*y*

_{1}≠b then slope of this line is

- .

When the centre of the circle is at the origin then the equation of the tangent line becomes

*x*_{1}*x*+*y*_{1}*y*=*r*^{2},

- .

## 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 centre forms a line of reflection symmetry and it has rotational symmetry around the centre 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 circle which is centred at the origin with radius 1 is called the unit circle.
- Thought of as a great circle of the unit sphere, it becomes the Riemannian 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 centre of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

### Chord

- Chords are equidistant from the centre of a circle if and only if they are equal in length.
- The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:
- A perpendicular line from the centre of a circle bisects the chord.
- The line segment (circular segment) through the centre 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 (see Thales' theorem).
- The diameter is the longest chord of the circle.
- If the intersection of any two chords divides one chord into lengths
*a*and*b*and divides the other chord into lengths*c*and*d*, then*ab*=*cd*. - If the intersection of any two perpendicular chords divides one chord into lengths
*a*and*b*and divides the other chord into lengths*c*and*d*, then*a*^{2}+*b*^{2}+*c*^{2}+*d*^{2}equals the square of the diameter. - The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.
^{[3]}^{:p.71}

### Sagitta

- The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc 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:

*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

*r*−

*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 (2

*r*−

*x*)

*x*= (

*y*/2)². Solving for

*r*, we find the required result.

### Tangent

- The line perpendicular drawn 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 centre 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.
- If a tangent at
*A*and a tangent at*B*intersect at the exterior point*P*, then denoting the center as*O*, the angles ∠*BOA*and ∠*BPA*are supplementary. - If
*AD*is tangent to the circle at*A*and if*AQ*is a chord of the circle, then ∠*DAQ*= arc(*AQ*).

### Theorems

See also: Power of a point

- The chord theorem states that if two chords,
*CD*and*EB*, intersect at*A*, then*CA*×*DA*=*EA*×*BA*. - 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*DC*^{2}=*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 one half the subtended angle on the opposite side of the chord (Tangent Chord Angle).
- If the angle subtended by the chord at the centre 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 secant-secant theorem.

### Inscribed angles

See also: Inscribed angle theorem

An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (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 is a right angle (since the central angle is 180 degrees).## Apollonius circle

Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant*ratio*(other than 1) of distances to two fixed foci, A and B. (The set of points where the distances are equal is the perpendicular bisector of A and B, a line.) That circle is sometimes said to be drawn

*about*two points

^{[4]}.

The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:

### Cross-ratios

A closely related property of circles involves the geometry of the cross-ratio 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 cross-ratio is equal to one:

*P*is a point on the Apollonius circle if and only if the cross-ratio [

*A*,

*B*;

*C*,

*P*] is on the unit circle in the complex plane.

### Generalized circles

See also: Generalized circle

If *C*is the midpoint of the segment

*AB*, then the collection of points

*P*satisfying the Apollonius condition

- (1)

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 a generalized circle of infinite radius.

## See also

## Notes

**^**Chronology for 30000 BC to 500 BC**^**Squaring the circle**^**Johnson, Roger A.,*Advanced Euclidean Geometry*, Dover Publ., 2007.**^**Harkness, James (1898).*Introduction to the theory of analytic functions*. London, New York: Macmillan and Co.. pp. 30. http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01680002.

## References

- Pedoe, Dan (1988).
*Geometry: a comprehensive course*. Dover. - "Circle" in The MacTutor History of Mathematics archive

## External links

Wikimedia Commons has media related to: Circle geometry |

Wikiquote has a collection of quotations related to: Circles |

- Circle (PlanetMath.org website)
- Weisstein, Eric W., "Circle" from MathWorld.
- 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
- Munching on Circles at cut-the-knot
- Ron Blond homepage - interactive applets
- calculate circumference and area with your own values
- MathAce's Circle article - has a good in-depth explanation of unit circles and transforming circular equations.
- Google Maps Circle Overlay Lets you add a circle to Google Maps

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