Difference between revisions of "Circle"
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== Traditional Definition == | == Traditional Definition == | ||
| − | A '''circle''' is defined as the [[set]] (or [[locus]]) of [[point]]s with an equal | + | A '''circle''' is defined as the [[set]] (or [[locus]]) of [[point]]s in a [[plane]] with an equal distance 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> | <center>[[Image:circle1.PNG]]</center> | ||
== Coordinate Definition == | == 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 | + | 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> | <center><math> (x-h)^2 + (y-k)^2 = r^2. </math></center> | ||
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== Area of a Circle == | == Area of a Circle == | ||
| − | The area of a circle is <math> \pi r^2 </math> where <math> \pi </math> is the mathematical constant [[pi]]. | + | The area of a circle is <math> \pi r^2 </math> where <math> \pi </math> is the mathematical constant [[pi]] and <math> r </math> is the radius. |
=== Archimedes' Proof === | === Archimedes' Proof === | ||
Revision as of 16:35, 30 June 2006
Contents
Traditional Definition
A circle is defined as the set (or locus) of points in a plane with an equal distance 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 and is the radius.
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.
