Circle

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.