Name:  Jay Wilkins

Title:  How many sides does a circle have?

Description:  This lesson investigates what happens as one allows the number of sides of a regular polygon to approach infinity.

How many sides does a circle have?

Polygons vs. Circles

A polygon is a closed geometric figure created by connecting line segments end to end so that no segment intersects any other segment in more than one point and every segment intersects exactly two other segments.  A regular polygon is a polygon whose sides and angles all have the same measure.
A circle is a set of points a given distance from a given point.
Look at the figure on the title page.  Is this figure a circle?

Answer:  NO.  In fact this is a polygon, a regular 30-gon.  How can this be?  It looks like a circle.  Let's investigate.  First let's look at regular polygons with 3 through 10 sides.

NPolygon[n_]:=Show[Graphics[{RGBColor
                  [Abs[N[Sin[n]]],Abs[N[Cos[n]]],
                  1-Abs[N[Sin[n]]]],
                Polygon[
														Table[{Cos[t],Sin[t]},{t,0,2Pi,2Pi/n}]]}],    
                  Graphics[Text[n"-gon",{0,1.1}]],
                  AspectRatio->Automatic, Axes->None]                  
																	Table[NPolygon[n],{n,3,10}]

What seems to be happening as the number of sides increase?

Answer:  The polygons begin to look more like a circle.  Let n get even bigger, say 20.

Table[NPolygon[n],{n,10,20}]

WOW!  That's almost a circle.  Let's look at a regular 30-gon.

NPolygon[30]

You see the title page figure is actually a polygon.  If we were to allow n to increase to infinity what would happen? Well, of course we cannot reach infinity but if we could we would have a circle.  So I guess we could say that a circle has an infinite number of sides?  Maybe.  But what is happening to the length of each side as the number of sides increase?  Are the sides still sides (remember, sides are defined to be line segments)?  Mmmmm!  Think about it!