Difference between revisions of "Circle"

Revision as of 01:13, 31 October 2006

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

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 21: / Line 21: @@
 <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>.
+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> \pi r^2 </math>.
 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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