# Circle: Wikis

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More interesting facts on Circle

# Encyclopedia

Circle illustration showing a radius, a diameter, the center and the circumference
Tycho crater, one of many examples of circles that arise in nature. NASA photo
.A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the center.^ (See sun symbol below the picture of the Eye of Horus ) A dot or point in the center of a circle symbolizes the blending of male and female forces.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

The common distance of the points of a circle from its center is called its radius.
.Circles are simple closed curves which divide the plane into two regions, an interior and an exterior.^ Sometimes this cross is seen with four additional "arms" dividing the circle into eight instead of four sections.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

^ Compare its two intersecting lines with Sun Sign 2 , its curved arms (following the shape of the circle) with Swastika 2, and its dark areas with the Iron Cross .
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

.In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure (known as the perimeter) or to the whole figure including its interior.^ The Caucasian Chalk Circle defines morality in terms of social 'use.'
• Bertolt Brecht (1898-1956) 20 September 2009 10:40 UTC www.csuchico.edu [Source type: FILTERED WITH BAYES]

^ We'll examine figures with circular boundaries like the vesica, and its use as art motifs.
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

However, in strict technical usage, "circle" refers to the perimeter while the interior of the circle is called a disk. The circumference of a circle is the perimeter of the circle (especially when referring to its length).
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.

## Further terminology

 Chord, secant, tangent, and diameter. Arc, sector, and segment
.The ""vertex"" of a circle is the point in which any line can cross.^ CIRCLE (quartered): The sacred circle filled with a cross, four equal lines pointing from the center to the spirits of the north, east, south, and west -- or to the basic element: earth, water, air (or wind), and fire.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

^ Compare its two intersecting lines with Sun Sign 2 , its curved arms (following the shape of the circle) with Swastika 2, and its dark areas with the Iron Cross .
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

.The diameter of a circle is the length of a line segment whose endpoints lie on the circle and which passes through the centre of the circle.^ Augustine described the nature of God as a circle whose centre was everywhere, and its circumference nowhere.
• EMERSON--"CIRCLES" hypertext 20 September 2009 10:40 UTC www.vcu.edu [Source type: Original source]

^ A great circle perpendicular to the plane of the ecliptic, passing through its poles.
• circle@Everything2.com 10 February 2010 13:27 UTC www.everything2.com [Source type: FILTERED WITH BAYES]

^ Circle of the sphere , a circle upon the surface of the sphere, called a great circle when its plane passes through the center of the sphere; in all other cases, a small circle .
• circle@Everything2.com 10 February 2010 13:27 UTC www.everything2.com [Source type: FILTERED WITH BAYES]

.This is the largest distance between any two points on the circle.^ Putting the clear transparency on the projector, I drew a large circle, then added two overlapping triangles (one pointing up, one pointing down), forming a hexagram.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

The diameter of a circle is twice its radius.
.The term " radius" can also refer to a line segment from the centre of a circle to its perimeter, and similarly the term "diameter" can refer to a line segment between two points on the perimeter which passes through the centre.^ Recall that the diameter of a circle will fit around the perimeter of the circle three times, plus a bit more, actually p times, where 3.1416.
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

^ Literature is a point outside of our hodiernal circle, through which a new one may be described.
• EMERSON--"CIRCLES" hypertext 20 September 2009 10:40 UTC www.vcu.edu [Source type: Original source]

^ Putting the clear transparency on the projector, I drew a large circle, then added two overlapping triangles (one pointing up, one pointing down), forming a hexagram.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

In this sense, the midpoint of a diameter is the centre and so it is composed of two radii.
.A chord of a circle is a line segment whose two endpoints lie on the circle.^ Compare its two intersecting lines with Sun Sign 2 , its curved arms (following the shape of the circle) with Swastika 2, and its dark areas with the Iron Cross .
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

.The diameter, passing through the circle's centre, is the largest chord in a circle.^ A great circle perpendicular to the plane of the ecliptic, passing through its poles.
• circle@Everything2.com 10 February 2010 13:27 UTC www.everything2.com [Source type: FILTERED WITH BAYES]

^ Circle of the sphere , a circle upon the surface of the sphere, called a great circle when its plane passes through the center of the sphere; in all other cases, a small circle .
• circle@Everything2.com 10 February 2010 13:27 UTC www.everything2.com [Source type: FILTERED WITH BAYES]

A tangent to a circle is a straight line that touches the circle at a single point. .A secant is an extended chord: a straight line cutting the circle at two points.^ The lines of the "A" often extend outside the circle.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

^ Putting the clear transparency on the projector, I drew a large circle, then added two overlapping triangles (one pointing up, one pointing down), forming a hexagram.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

^ Compare its two intersecting lines with Sun Sign 2 , its curved arms (following the shape of the circle) with Swastika 2, and its dark areas with the Iron Cross .
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

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 compass in this 13th century manuscript is a symbol of God's act of Creation.
^ This was symbolized in art by God holding a pair of compasses, a common motif in the Middle Ages.
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

Notice also the circular shape of the halo
.The circle has been known since before the beginning of recorded history.^ To most people, this is true enough, but to those paying attention, Circle has been making some of the most original modern rock music since the beginning of the ‘90s.
• CIRCLE on MySpace Music - Free Streaming MP3s, Pictures & Music Downloads 10 February 2010 13:27 UTC www.myspace.com [Source type: FILTERED WITH BAYES]

.It is the basis for the wheel, which, with related inventions such as gears, makes much of modern civilization possible.^ Character makes an overpowering present; a cheerful, determined hour, which fortifies all the company, by making them see that much is possible and excellent that was not thought of.
• EMERSON--"CIRCLES" hypertext 20 September 2009 10:40 UTC www.vcu.edu [Source type: Original source]

In mathematics, the study of the circle has helped inspire the development of geometry 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.^ Circles can be " called "/invoked/set up in many different ways, but the common standard is to call each of the elements (with calling the watchtowers being the most used there), then bind the four energies together.
• circle@Everything2.com 10 February 2010 13:27 UTC www.everything2.com [Source type: FILTERED WITH BAYES]

^ I believe this plan will likely be successful as there are many callers that cannot control the phone networks of their friends and family members.
• Alltel My Circle 10 February 2010 13:27 UTC blog.tmcnet.com [Source type: FILTERED WITH BAYES]

[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 as an approximate value of π.[1]
• .
• 300 BC – Book 3 of Euclid's Elements deals with the properties of circles.
• 1880 – Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle.^ Recall from our unit on Egypt we said that the problem of squaring the circle is one of constructing, using only compass and straightedge; .
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

[2]

## Analytic results

### Length of circumference

.The ratio of a circle's circumference to its diameter is π (pi), a constant that takes the same value (approximately 3.141592654) for all circles.^ Formulae vital to the geometric aspect of a circle : Circumfrence : c = {pi} * d Area : a = {pi} * r^2 Diameter : d = 2r Arc Length - The distance from one point on the circumference to another point on the circumference traveling along the edge of the circle.
• circle@Everything2.com 10 February 2010 13:27 UTC www.everything2.com [Source type: FILTERED WITH BAYES]

Thus the length of the circumference (c) is related to the radius (r) by
$c = 2 \pi r\,$
or equivalently to the diameter (d) by
$c = \pi d.\,$

### Area enclosed

Area of the circle = π × area of the shaded square
The area enclosed by a circle is π multiplied by the radius squared:
$Area = \pi r^2.\,$
Equivalently, denoting diameter by d,
$Area = \frac{\pi d^2}{4} \approx 0{.}7854 \cdot d^2,$
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.^ Compare its two intersecting lines with Sun Sign 2 , its curved arms (following the shape of the circle) with Swastika 2, and its dark areas with the Iron Cross .
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.

### Equations

#### Cartesian coordinates

Circle of radius r = 1, center (a, b) = (1.2, -0.5)
In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that
$\left(x - a \right)^2 + \left( y - b \right)^2=r^2.$
.This equation of the circle follows from the Pythagorean theorem 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.^ Putting the clear transparency on the projector, I drew a large circle, then added two overlapping triangles (one pointing up, one pointing down), forming a hexagram.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

If the circle is centered at the origin (0, 0), then the equation simplifies to
$x^2 + y^2 = r^2. \!\$
The equation can be written in parametric form using the trigonometric functions sine and cosine as
$x = a+r\,\cos t,\,\!$
$y = b+r\,\sin t\,\!$
where t is a parametric variable, interpreted geometrically as the angle that the ray from the origin to (xy) makes with the x-axis. Alternatively, a rational parametrization of the circle is:
$x = a + r \frac{1-t^2}{1+t^2}$
$y = b + r \frac{2t}{1+t^2}.$
In homogeneous coordinates each conic section with equation of a circle is of the form
$\ ax^2+ay^2+2b_1xz+2b_2yz+cz^2 = 0.$
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.

#### Polar coordinates

In polar coordinates the equation of a circle is:
$r^2 - 2 r r_0 \cos( heta - \varphi) + r_0^2 = a^2\,$
where a is the radius of the circle, r0 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. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes
$r = 2 a\cos( heta - \varphi)$.
In the general case, the equation can be solved for r, giving
$r = r_0 \cos( heta - \varphi) + \sqrt{a^2 - r_0^2 \sin^2( heta - \varphi)}$,
the solution with a minus sign in front of the square root giving the same curve.

#### Complex plane

In the complex plane, a circle with a center at c and radius (r) has the equation $|z-c|^2 = r^2\,$. In parametric form this can be written z = reit + c.
The slightly generalized equation $pz\overline{z} + gz + \overline{gz} = q$ for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with $p = 1,\ g=\overline{c},\ q=r^2-|c|^2$, since $|z-c|^2 = z\overline{z}-\overline{c}z-c\overline{z}+c\overline{c}$. 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 the circle is perpendicular to the diameter passing through P.^ Literature is a point outside of our hodiernal circle, through which a new one may be described.
• EMERSON--"CIRCLES" hypertext 20 September 2009 10:40 UTC www.vcu.edu [Source type: Original source]

If P = (x1, y1) and the circle has center (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1a)x+(y1b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is
(x1a)x + (y1b)y = (x1a)x1 + (y1b)y1
or
(x1a)(xa) + (y1b)(yb) = r2.
If y1≠b then slope of this line is
$\frac{dy}{dx} = -\frac{x_1-a}{y_1-b}$.
This can also be found using implicit differentiation.
When the center of the circle is at the origin then the equation of the tangent line becomes
x1x + y1y = r2,
and its slope is
$\frac{dy}{dx} = -\frac{x_1}{y_1}$.

## Properties

.
• The circle is the shape with the largest area for a given length of perimeter.^ Compare its two intersecting lines with Sun Sign 2 , its curved arms (following the shape of the circle) with Swastika 2, and its dark areas with the Iron Cross .
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

(See Isoperimetric inequality.)
• .
• The circle is a highly symmetric shape: every line through the center forms a line of reflection symmetry and it has rotational symmetry around the center for every angle.^ So every day you wish to change your circle, just drop into your nearest Alltel store and wait in line.
• Alltel My Circle 10 February 2010 13:27 UTC blog.tmcnet.com [Source type: FILTERED WITH BAYES]

^ Compare its two intersecting lines with Sun Sign 2 , its curved arms (following the shape of the circle) with Swastika 2, and its dark areas with the Iron Cross .
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.
• All circles are similar.
• The circle which is centered at the origin with radius 1 is called the unit circle.^ In that same unit we also saw that a circle whose radius is the pyramid height .
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

.
• Through any three points, not all on the same line, there lies a unique circle.^ In that same unit we also saw that a circle whose radius is the pyramid height .
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

^ But when you get to the top, the points all come together, and there the eye of God opens."
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

.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 sun symbol below the picture of the Eye of Horus ) A dot or point in the center of a circle symbolizes the blending of male and female forces.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

See circumcircle.

### Chord

.
• 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 (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.^ Only the Chalk Circle can decide and in the end a great lesson is learned regarding the people of Grusinia and the valley they are fighting over" [ stress added].
• Bertolt Brecht (1898-1956) 20 September 2009 10:40 UTC www.csuchico.edu [Source type: FILTERED WITH BAYES]

• 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

• The sagitta (also known as the versine) 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:
$r=\frac{y^2}{8x}+ \frac{x}{2}.$
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.^ Reply i wanted to know since i ahve an prepaid alltel phone can i still put my friends in my circle if so how much do it cost?
• Alltel My Circle 10 February 2010 13:27 UTC blog.tmcnet.com [Source type: FILTERED WITH BAYES]

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.^ We will continue to delight in the cross, while recognizing that others use the same image to represent their dark forces.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

Solving for r, we find the required result.

### Tangent

• 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

.
Secant-secant theorem
• 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 DC2 = DG×DE.^ Putting the clear transparency on the projector, I drew a large circle, then added two overlapping triangles (one pointing up, one pointing down), forming a hexagram.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

^ (See sun symbol below the picture of the Eye of Horus ) A dot or point in the center of a circle symbolizes the blending of male and female forces.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

(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 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).^ One of two prop boxes for The Caucasian Chalk Circle .
• Bertolt Brecht (1898-1956) 20 September 2009 10:40 UTC www.csuchico.edu [Source type: FILTERED WITH BAYES]

^ Do you wonder why PBS would choose a similar cross (right side) to represent the Quartered Circle of the earth-centered religions of Aborigenes around the world?
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

^ "The cross inscribed in a circle mediates between the square and the circle," emphasizing the "joining of heaven and earth....
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

This is the secant-secant theorem.

### Inscribed angles

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' definition of a circle: d1/d2 constant
.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.^ CIRCLE (quartered): The sacred circle filled with a cross, four equal lines pointing from the center to the spirits of the north, east, south, and west -- or to the basic element: earth, water, air (or wind), and fire.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

^ Reply will you email me show hoe to set up circle please .
• Alltel My Circle 10 February 2010 13:27 UTC blog.tmcnet.com [Source type: FILTERED WITH BAYES]

) .That circle is sometimes said to be drawn about two points[3].^ Putting the clear transparency on the projector, I drew a large circle, then added two overlapping triangles (one pointing up, one pointing down), forming a hexagram.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

^ The speculation about why a stone circle was flattened was to make its perimeter an integral multiple of the radius drawn to their circular part of its perimeter.
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:
$\frac{AP}{BP} = \frac{AC}{BC}.$
Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to $180^{\circ}$, the angle CPD is exactly $90^{\circ}$, 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.^ Putting the clear transparency on the projector, I drew a large circle, then added two overlapping triangles (one pointing up, one pointing down), forming a hexagram.
• Symbols and their meaning 10 February 2010 13:27 UTC www.crossroad.to [Source type: FILTERED WITH BAYES]

### 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:
$|[A,B;C,P]| = 1.\$
.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 in the complex plane.^ Recall from our unit on Egypt we said that the problem of squaring the circle is one of constructing, using only compass and straightedge; .
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

^ The Greek cross within a circle (cruciform nimbus) is used only when portraying Christ.
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

^ "Instinctive compassion is a major subject of Brecht's last major dramatic work, The Caucasian Chalk Circle , which was written in 1944-45 while Brecht lived in exile in the United States.
• Bertolt Brecht (1898-1956) 20 September 2009 10:40 UTC www.csuchico.edu [Source type: FILTERED WITH BAYES]

### Generalized circles

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition
$\frac{|AP|}{|BP|} = \frac{|AC|}{|BC|}$   (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 a generalized circle of infinite radius.

## Notes

1. ^ Chronology for 30000 BC to 500 BC
2. ^ Squaring the circle
3. ^ Harkness, James (1898). Introduction to the theory of analytic functions. London, New York: Macmillan and Co.. pp. 30.

# Travel guide

Up to date as of January 14, 2010

### From Wikitravel

.Circle is an old gold rush settlement on the Yukon River in a remote section of Interior Alaska.^ Circle (also known as Circle City) was established in 1893 as a supply point for goods shipped up the Yukon River and then overland to the gold mining camps.
• Alaska Division of Community and Regional Affairs 10 February 2010 13:27 UTC www.commerce.state.ak.us [Source type: FILTERED WITH BAYES]

^ By 1896, before the Klondike gold rush, Circle was the largest mining town on the Yukon, with a population of 700.
• Alaska Division of Community and Regional Affairs 10 February 2010 13:27 UTC www.commerce.state.ak.us [Source type: FILTERED WITH BAYES]

^ FRIES & ONION RINGS Arctic Circle's original fry sauce remains the favorite compliment to the restaurants’ Yukon Gold Fries and real onion rings.
• Arctic Circle Restaurants Inc. -"Where the Good Stuff Is" 20 September 2009 10:40 UTC www.arcticcirclerest.com [Source type: General]

.The gold rush is over, and the population today is small and mostly indigenous.^ By 1896, before the Klondike gold rush, Circle was the largest mining town on the Yukon, with a population of 700.
• Alaska Division of Community and Regional Affairs 10 February 2010 13:27 UTC www.commerce.state.ak.us [Source type: FILTERED WITH BAYES]

.It is located 50 miles south of the Arctic Circle, but was named "Circle" because miners thought it was at the Arctic Circle.^ Early miners believed the town was located on the Arctic Circle, and named it Circle.
• Alaska Division of Community and Regional Affairs 10 February 2010 13:27 UTC www.commerce.state.ak.us [Source type: FILTERED WITH BAYES]

^ Comfort Inn & Suites Hotel Circle South Hotel Circle Inn and Suites is located in the heart of San Diego in what is known as Mission Valley.

^ Ramada Plaza Hotel Circle South Centrally located on famous Hotel Circle, you will be able to sleep between the Zoo and Sea World.

• Visit Circle Hot Springs

# 1911 encyclopedia

Up to date as of January 14, 2010
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# Wiktionary

Up to date as of January 15, 2010
(Redirected to circle article)

## English

Wikipedia has an article on:
A circle

### Noun

 Singular circle Plural circles
circle (plural circles)
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1. (geometry): A two-dimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from another point.^ In symbolic logic, this figure represents the intersection of two sets, a good symbol for the intersection of Art & Geometry.
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

^ Another geometric figure made from circular arcs is the vesica or mandorla , a very common geometric figure in art history.
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

The set of all points (x, y) such that (x − 1)2 + y2 = r2 is a circle of radius r around the point (1, 0).
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2. A two-dimensional geometric figure, a disk, consisting of the set of all those points of a plane at a distance less than or equal to a fixed distance from another point.
3. Any thin three-dimensional equivalent of the geometric figures.^ In symbolic logic, this figure represents the intersection of two sets, a good symbol for the intersection of Art & Geometry.
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

^ Another geometric figure made from circular arcs is the vesica or mandorla , a very common geometric figure in art history.
• Geometry in Art & Architecture Unit 9 10 February 2010 13:27 UTC www.dartmouth.edu [Source type: FILTERED WITH BAYES]

Put on your dunce-cap and sit down on that circle.
4. A curve that more or less forms part or all of a circle.
move in a circle
5. Orbit.
6. A specific group of persons.
inner circle
circle of friends
7. (cricket) A line comprising two semicircles of 30 yards radius centred on the wickets joined by straight lines parallel to the pitch used to enforce field restrictions in a one-day match.

#### Synonyms

• (two-dimensional outline geometric figure): coil (not in mathematical use), ring (not in mathematical use), loop (not in mathematical use)
• (two-dimensional solid geometric figure): disc/disk (in mathematical and general use), round (not in mathematical use; UK & Commonwealth only)
• (curve): arc, curve
• (orbit): orbit

#### Translations

.The translations below need to be checked and inserted above into the appropriate translation tables, removing any numbers.^ Reply Need to check numbers on my circle.My number is 218-790-6103.
• Alltel My Circle 10 February 2010 13:27 UTC blog.tmcnet.com [Source type: FILTERED WITH BAYES]

Numbers do not necessarily match those in definitions. See instructions at Help:How to check translations.

### Verb

 Infinitive to circle Third person singular circles Simple past circled Past participle circled Present participle circling
to circle (third-person singular simple present circles, present participle circling, simple past and past participle circled)
1. (transitive) To travel around along a curved path.
2. (transitive) To surround.
3. (transitive) To place or mark a circle around.
Circle the jobs that you are interested in applying for.
4. (intransitive) To travel in circles.

#### Translations

.The translations below need to be checked and inserted above into the appropriate translation tables, removing any numbers.^ Reply Need to check numbers on my circle.My number is 218-790-6103.
• Alltel My Circle 10 February 2010 13:27 UTC blog.tmcnet.com [Source type: FILTERED WITH BAYES]

Numbers do not necessarily match those in definitions. See instructions at Help:How to check translations.

# Simple English

A circle is a round two-dimensional shape, such as the letter o.

The centre of a circle is the point in the very middle.

The radius of a circle is a line from the centre of the circle to a point on the side.

All points on the circle are at the same distance from the centre. In other words, the radius is the same length all the way around the circle. Mathematicians use the letter r for the length of a circle's radius.

The diameter (meaning "all the way across") of a circle is a straight line that goes from one side to the opposite and right through the centre. Mathematicians use the letter d for the length of this line.

The diameter of a circle is equal to twice its radius (d equals 2 times r).



d = 2\ r

The circumference (meaning "all the way around") of a circle is line that goes around the circle. Mathematicians use the letter c for the length of this line.

The number π (written as the Greek letter pi) is a very useful number. It is the length of the circumference divided by the length of the diameter (π equals c divided by d). The number π is equal to about 227 or 3.14159.

 $\pi = \frac\left\{c\right\}\left\{d\right\}$ $\therefore$ $c = 2\pi \, r$

The area, a, inside a circle is equal to the radius multiplied by itself, then multiplied by π (a equals π times (r times r)).

$a = \pi \, r^2$

## Calculating π

π can be empirically measured by drawing a large circle, then measuring its diameter and circumference, since the circumference of a circle is always π times its diameter.

π can also be calculated using purely mathematical methods. Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. However, some are quite simple, such as this form of the Gregory-Leibniz series:

$\pi = \frac\left\{4\right\}\left\{1\right\}-\frac\left\{4\right\}\left\{3\right\}+\frac\left\{4\right\}\left\{5\right\}-\frac\left\{4\right\}\left\{7\right\}+\frac\left\{4\right\}\left\{9\right\}-\frac\left\{4\right\}\left\{11\right\}\cdots$

While that series is easy to write and calculate, it is not immediately obvious why it yields π. A more intuitive approach is to draw an imaginary circle of radius r centered at the origin. Then any point (x,y) whose distance d from the origin is less than r, as given by the pythagorean theorem, will be inside the circle:

$d = \sqrt\left\{x^2 + y^2\right\}$

Finding a collection of points inside the circle allows the circle's area A to be approximated. For example, by using integer coordinate points for a big r. Since the area A of a circle is π times the radius squared, π can be approximated by using:

$\pi = \frac\left\{A\right\}\left\{r^2\right\}$

## Other websites

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# Citable sentences

Up to date as of December 19, 2010

Here are sentences from other pages on Circle, which are similar to those in the above article.