Finding the Circumference

Date: 11/27/97 at 13:30:48
From: Shae Godfrey
Subject: Circumference

Would you be able to tell me how to work out the circumference of a 
circle?

Date: 11/27/97 at 17:49:10
From: Doctor Luis
Subject: Re: Circumference

Basically, the "circumference" of a circle is the distance around it.
There is a way to calculate the circumference if you know another
geometric length called the radius.

Here's a diagram of a circle (sort of). Okay, it's not a circle, but
it's as close as I can get with these characters. :)

So suppose it's a circle...

        *   *      Do you see the "+"? It represents a special point
     *         *   on the circle, called the "center." The reason it's
    *   r       *  a special point is that if you make a line from
    *-----+     *  the center to a point on the circle, that line
    *           *  is going to have the same length no matter what
     *         *   point on the circle you choose. Those special lines
        *   *      are called "radii" (pronounce the "ii" as an "e"
                   followed by an "i"). If you're only talking about
one of these lines then it is called "radius". Here, on the diagram, I 
called it the radius "r".

 (Notice that there are an infinite number of these "radii" but
 their lengths are the same)

There is another special line that is associated with the circle, and 
mathematicians have called it the "diameter". It is simply a line 
segment from one point on the circle to another point on the circle 
that also passes through the center of the circle. It looks something 
like this,

        *  *      If you look carefully, you can see that the diameter
     *        *    (I called it D on the diagram) is in reality one
    *    D     *   radius from the center to a point on the circle,
    *-----+----*   and another radius from the center to the OPPOSITE
    *          *   point on the circle. So you see, the length of
     *        *    the diameter is twice the length of the radius.
        *  * 

Now, what about the circumference? Well, that's just the curved line
around the circle. This means that if you cut your circle at one point 
and straighten it so that it becomes a straight line, the length of 
that line is going to be the length of the circumference.

  |<---    Circumference     --->|
  --------------------------------

It turns out that circles have a very curious property. If you take 
the diameter of ANY circle (no matter what size), the number of 
diameters that fit in the circumference will be the same for ANY 
circle (try to verify it yourself with two pieces of string). Actually 
the diameter will fit about three times into the circumference, but 
not EXACTLY. There will be a small part of the circumference left.

    diameter
  ----------

    circumference
  --------------------------------

    three diameters
  ------------------------------

Mathematically speaking, we say that the ratio of the circumference to 
the diameter is constant (the same number) for all circles. This  
constant is very famous in mathematics, and it appears unexpectedly in 
a lot of theorems and equations. Even the Egyptians knew about it 
thousands of years ago (they said it was 22/7 although we know it's a 
little less than that). The Greeks also knew about this constant and 
calculated it to better accuracy using geometrical methods. 
Mathematicians like to call this constant "pi" (pronounced like 
"pie"), after the Greek letter "p". 

This number pi is a very interesting number that people have studied 
extensively. It is an irrational number. Essentially, that means you 
cannot write it as a ratio (you can't get pi by dividing two 
integers). The implication of that is that we will never know the 
exact value of pi - we will always be off the true value. We can only 
get increasingly accurate estimates of what pi is, and people have 
written programs and found pi to millions of decimal places on the 
most powerful and fastest computers. (for most purposes we only need 
fewer than ten decimal places).

Well, how does all of this help us calculate the circumference of a 
circle? Simple, you know that the circumference divided by the 
diameter is the number pi for any circle, and so

   pi = Circumference / Diameter

 or

   circumference = pi * diameter

So, if you know the diameter of the circle you can get the 
circumference just by multiplying by pi (which is approximately 
3.14159... ) .

Remember that,

   diameter = 2 * radius

and so

   circumference = pi * (2 * radius)
                 = 2 pi * radius

  or simply

    C = 2 pi r


This is the way you'll find it in most books, in terms of the radius. 

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