Circle: Difference between revisions

Revision as of 16:08, 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.

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.

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:

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.

@@ Line 1: / Line 1: @@
-==Definition==
+== Traditional Definition ==
-*The [[locus]] of all points in a [[plane]] with distance <math>\displaystyle {r}</math>(radius) from center <math>{O}</math>
+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]].
+<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>
+== 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==
@@ Line 17: / Line 44: @@
 == See Also ==
-*[[Pi]]
+* [[Dandelin Sphere]]s
-*[[Power of a point]]
+* [[Geometry]]
-*[[Law of sines]]
+* [[Pi]]
-*[[Inversion]]
+* [[Power of a point]]
-*[[Homothecy]]
+* [[Inversion]]
+* [[Homothecy]]

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