Difference between revisions of "Circle"
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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> | + | 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]]. | Archimedes also came up with a brilliant proof of the area of a circle by using the [[proof]] technique of [[reductio ad absurdum]]. |
Revision as of 01:13, 31 October 2006
Contents
[hide]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.