# Circle

## 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.