Polygons, Infinite Sides, and Circles

Date: 04/03/97 at 20:33:18
From: Linda Solomon
Subject: Polygons and Circles

Dear Dr. Math,

A polygon is by definition a closed figure with straight line 
segments.  A regular polygon is a polygon that has congruent sides and 
congruent angles. 

My math teacher said that if you take a regular polygon and put on 
an infinite number of sides it would become a circle.  I don't agree. 

Josh Solomon, New Rochelle, New York  

Date: 04/04/97 at 10:28:01
From: Doctor Mitteldorf
Subject: Re: Polygons and Circles

Dear Josh,

Talking about infinity is always tricky.  There's a way in which 
what your teacher said is true, and a way in which it isn't.  

Here's a useful idea that comes up in mathematics you'll learn in 
HS and college. It's the idea of a limit. Say you can't calculate 
something about x, but you can calculate it for other numbers 
besides x. Well, choose a number y that's close to x and calculate it.  
Then calculate the thing for y. Try letting y get a little closer to 
x, and calculate it again. Keep going. See if the things you're 
calculating seem to be "homing in" on something - that is, they get
closer and closer, but never quite reach it.

This is a useful way to talk about infinity because if your x is 
infinity, there's nothing you can calculate about x directly, but you 
can calculate for y and let y get larger and larger - that is, 
"closer" to infinity.  Then see if the values seem to be homing in on 
something.

Why don't you try doing this with the perimeter of the polygon.  
Try writing down the perimeter for a triangle, a square, a pentagon, 
hexagon...  You may need some help from your teacher or parent 
calculating these perimeters.

If you keep going, you'll find something funny.  The number of sides 
keeps going up and up, and the perimeter goes up and up.  But the 
perimeter goes up much slower than the number of sides is going up.  
After a while, it will seem that the perimeter is just grinding to 
a halt - getting closer and closer to a value that it just can't seem 
to get past.  That value will be the circumference of a circle!

The same thing works for areas. The areas of all those polygons gets 
closer and closer to the area of a circle.  

That's really what your math teacher meant when he said that if you 
took "a regular polygon and put on an infinite number of sides it 
would become a circle".  He was using a shorthand that he learned a 
long time ago. He didn't mean literally that it would BE a circle, but 
rather it would be more and more like a circle. 

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/