Explorations of the Circle

Date: 02/09/99 at 20:59:03
From: Garrett Hill
Subject: Math

I want to know what pi means. I am studying about circles, like about 
chords and radius. I would like to know the definition of circle and 
whether there is something else to use instead of radius, chord, 
diameter, and center.

Sincerely,
Garrett Hill

Date: 02/09/99 at 23:16:41
From: Doctor Kate
Subject: Re: Math

Garrett:

You're asking some of the same questions that people have asked for 
thousands of years. They're very interesting questions. A long time 
ago the Greeks wondered what pi was, and a lot of people tried to find 
out, by trying to draw circles and squares with the same area. Even in 
this century, people have claimed that they've discovered that pi is 
22 divided by 7.

Pi is the ratio of the circumference of a circle to its diameter. That 
is, if you have a string the length of the circumference, Pi is how 
many times it will cover the diameter. It's a little bit more than 3. 
But Pi has all sorts of other strange properties. It's a number that 
you can never write all of. (That's why we just say "Pi" instead of 
writing it down.) If you start to write Pi, it looks like 
3.141592653589 but that's only the beginning of it. It goes on forever.

But what's the definition of a circle? Well, here's one interesting 
way to think of it:

Put a thumbtack or a nail in a tabletop and tie a string to it. Tie the 
other end of the string to a pencil. Now pull the pencil as far away 
from the nail as you can and put the point on the table. The string 
will stop you from pulling too far. Now if you move the pencil while 
keeping the string pulled tight, you can move it around the nail and it 
will draw a circle. You should try this (ask before you put a nail or a 
thumbtack into someone's table, though.)

Now, if we think of drawing a circle this way, we will realise that the 
pencil is always a certain distance from the nail - the distance is the 
length of the string. So you could define a circle this way: It's made 
up of all the points that are the same distance from the centre point.

If you draw a point on a piece of paper (the centre point) and then use 
your ruler to draw other points (whichever ones you like) that are 
exactly 3 centimetres from the centre point, you'll find that you start 
to get a circle once you've drawn lots of these points. If you could 
draw enough points, you could "fill up" this whole circle.

There are lots of ways to define a circle, but I like that one myself.

Now for your last question: What else is there besides radius, chord, 
diameter and centre? Well, you can make up anything you like. You could 
draw triangles and squares inside circles, or put circles inside 
squares. You could make a circle inside another one with a radius half 
of the big one. You could make spirals instead of circles. You could 
try to figure out what the area of a circle is. There are always lots 
of things to explore.

Here's something to try once you've tried making a circle with a nail 
and string: try putting TWO nails in a board or a tabletop, and then 
tying a string in a loop around the two nails. Then put your pencil in 
the string and pull it tight (so you have a triangle made of two nails 
and a pencil in a loop of string). Now try moving the pencil while 
keeping the string tight. What shape do you draw?

Have fun!

- Doctor Kate, The Math Forum
  http://mathforum.org/dr.math/