Difference between revisions of "Circle"

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==Definition==
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== Traditional Definition ==
*The [[locus]] of all points in a [[plane]] with distance <math>\displaystyle {r}</math>(radius) from center <math>{O}</math>
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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]].
*A cross section of a [[cone]] or [[cylinder]] parallel to the base. For more on this, see [[Dandelin Spheres]], a commonly used method of deriving all [[Conic sections]].
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<center>[[Image:circle1.PNG]]</center>
* All x, y such that <math>(x - a)^{2} + (y - b)^{2} = r^{2} </math> for a circle centered at <math>(a,b)</math> with radius <math>{r}</math>
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== Coordinate Definition ==
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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
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<center><math> (x-h)^2 + (y-k)^2 = r^2. </math></center>
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'''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.
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<center>[[Image:Circlecoordinate1.PNG]]</center>
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== Area of a Circle ==
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The area of a circle is <math> \pi r^2 </math> where <math> \pi </math> is the mathematical constant [[pi]].
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=== Archimedes' Proof ===
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We shall explore two of the Greek [[mathematician]] [[Archimedes]] demonstrations of the area of a circle. The first is much more intuitive.
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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:
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<center>[[Image:Pizzawedges2.PNG]]</center>
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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>.
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Archimedes also came up with a brilliant proof of the area of a circle by using the [[proof]] technique of [[reductio ad absurdum]].
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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.
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'''Case 1:''' The circle's area is greater than the triangle's area.
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''This proof needs to be finished.''
  
 
==Formulas==
 
==Formulas==
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== See Also ==
 
== See Also ==
*[[Pi]]
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* [[Dandelin Sphere]]s
*[[Power of a point]]
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* [[Geometry]]
*[[Law of sines]]
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* [[Pi]]
*[[Inversion]]
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* [[Power of a point]]
*[[Homothecy]]
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* [[Inversion]]
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* [[Homothecy]]

Revision as of 17:08, 30 June 2006

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.

Circle1.PNG

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, $r$, and center $(h,k)$. We know that each point, $(x,y)$, on the circle which we want to identify is a distance $r$ from $(h,k)$. Using the distance formula, this gives $\sqrt{(x-h)^2 + (y-k)^2} = r$ which is more commonly written as

$(x-h)^2 + (y-k)^2 = r^2.$

Example: The equation $(x-3)^2 + (y+6)^2 = 25$ represents the circle with center $(3,-6)$ and radius 5 units.

Circlecoordinate1.PNG

Area of a Circle

The area of a circle is $\pi r^2$ where $\pi$ 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:

Pizzawedges2.PNG

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 $r$ and width $\pi r$ thus making its area $2\pi r$.

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 $r$ and circumference $C$ had an area equivalent to the area of a right triangle with base $C$ and height $r$. First let the area of the circle be $A$ and the area of the triangle be $T$. 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 $\displaystyle \pi r^2$
  • circumference $\displaystyle 2\pi r$

Other Properties

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

Practice Problems

See Also

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