# Difference between revisions of "Circle"

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− | ==Definition== | + | == Traditional Definition == |

− | + | A '''circle''' is defined as the [[set]] (or [[locus]]) of [[point]]s with an equal distant from a fixed point. The fixed point is called the [[center]] and the distance from the center to a point on the circle is called the [[radius]]. | |

− | + | <center>[[Image:circle1.PNG]]</center> | |

− | + | ||

+ | == Coordinate Definition == | ||

+ | Using the traditional definition of a circle, we can find the general form of the equation of a circle on the coordinate plane given its radius, <math> r </math>, and center <math> (h,k) </math>. We know that each point, <math> (x,y) </math>, on the circle which we want to identify is a distance <math> r </math> from <math> (h,k) </math>. Using the distance formula, this gives <math> \sqrt{(x-h)^2 + (y-k)^2} = r </math> which is more commonly written as | ||

+ | |||

+ | <center><math> (x-h)^2 + (y-k)^2 = r^2. </math></center> | ||

+ | |||

+ | '''Example:''' The equation <math> (x-3)^2 + (y+6)^2 = 25 </math> represents the circle with center <math> (3,-6) </math> and radius 5 units. | ||

+ | <center>[[Image:Circlecoordinate1.PNG]]</center> | ||

+ | |||

+ | == Area of a Circle == | ||

+ | The area of a circle is <math> \pi r^2 </math> where <math> \pi </math> is the mathematical constant [[pi]]. | ||

+ | |||

+ | === Archimedes' Proof === | ||

+ | We shall explore two of the Greek [[mathematician]] [[Archimedes]] demonstrations of the area of a circle. The first is much more intuitive. | ||

+ | |||

+ | Archimedes envisioned cutting a circle up into many little wedges (think of slices of pizza). Then these wedges were placed side by side as shown below: | ||

+ | |||

+ | <center>[[Image:Pizzawedges2.PNG]]</center> | ||

+ | |||

+ | As these slices are made infinitely thin, the little green arcs in the diagram will become the blue line and the figure will approach the shape of a rectangle with length <math> r </math> and width <math> \pi r </math> thus making its area <math> 2\pi r </math>. | ||

+ | |||

+ | Archimedes also came up with a brilliant proof of the area of a circle by using the [[proof]] technique of [[reductio ad absurdum]]. | ||

+ | |||

+ | Archimedes' actual claim was that a circle with radius <math> r </math> and circumference <math> C </math> had an area equivalent to the area of a [[right triangle]] with base <math> C </math> and height <math> r </math>. First let the area of the circle be <math> A </math> and the area of the triangle be <math> T </math>. We have three cases then. | ||

+ | |||

+ | '''Case 1:''' The circle's area is greater than the triangle's area. | ||

+ | |||

+ | ''This proof needs to be finished.'' | ||

==Formulas== | ==Formulas== | ||

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== See Also == | == See Also == | ||

− | *[[ | + | * [[Dandelin Sphere]]s |

− | *[[ | + | * [[Geometry]] |

− | *[[ | + | * [[Pi]] |

− | *[[Inversion]] | + | * [[Power of a point]] |

− | *[[Homothecy]] | + | * [[Inversion]] |

+ | * [[Homothecy]] |

## Revision as of 17:08, 30 June 2006

## Contents

## Traditional Definition

A **circle** is defined as the set (or locus) of points with an equal distant from a fixed point. The fixed point is called the center and the distance from the center to a point on the circle is called the radius.

## Coordinate Definition

Using the traditional definition of a circle, we can find the general form of the equation of a circle on the coordinate plane given its radius, , and center . We know that each point, , on the circle which we want to identify is a distance from . Using the distance formula, this gives which is more commonly written as

**Example:** The equation represents the circle with center and radius 5 units.

## Area of a Circle

The area of a circle is where is the mathematical constant pi.

### Archimedes' Proof

We shall explore two of the Greek mathematician Archimedes demonstrations of the area of a circle. The first is much more intuitive.

Archimedes envisioned cutting a circle up into many little wedges (think of slices of pizza). Then these wedges were placed side by side as shown below:

As these slices are made infinitely thin, the little green arcs in the diagram will become the blue line and the figure will approach the shape of a rectangle with length and width thus making its area .

Archimedes also came up with a brilliant proof of the area of a circle by using the proof technique of reductio ad absurdum.

Archimedes' actual claim was that a circle with radius and circumference had an area equivalent to the area of a right triangle with base and height . First let the area of the circle be and the area of the triangle be . We have three cases then.

**Case 1:** The circle's area is greater than the triangle's area.

*This proof needs to be finished.*

## Formulas

**Area****circumference**

## Other Properties

- awaiting diagrams to add stuff on inscribed angles + tangents.