An Introduction to Pi and Its Use in Circle Problems

Date: 04/28/2004 at 22:46:36
From: josh
Subject: i have lots of trouble with pi problems will you please help

I don't understand problems that involve pi.  For example, if the
circumference of a circle is 66 mm, what is the radius?  I just don't
understand what pi is and how you use it.

Date: 04/29/2004 at 09:38:19
From: Doctor Ian
Subject: Re: i have lots of trouble with pi problems will you please help

Hi Josh,

One of the most common places that we run into pi is in dealing with circles.
For any circle, there are a number of things we can measure or calculate.
These include:

  1.  The circumference

      This is the distance around the circle.  That is, if we
      make a mark on the circle, then go around the circle until
      we get back to the mark, the distance we travel is the 
      circumference.

  2.  The radius

      Every point on a circle is the same distance from the
      center of the circle.  This distance is called the radius
      of the circle.  That is, if we start at the center and 
      move to any point on the circle, the distance we travel
      is the radius. 

  3.  The diameter

      The diameter is just twice the radius.  Another way to think
      about it is that if we pick any point on a circle, the distance
      to the point that is farthest away--directly on the opposite
      side of the center--is the diameter.  

  4.  The area

      Area is different than distance, in that it's two-dimensional,
      where distance is one-dimensional.  If you're not clear on the
      difference between the two, take a look at 

        Area and Perimeter
          http://mathforum.org/library/drmath/view/57652.html 

Okay, so what does pi have to do with all this?  Well, if we pick
_any_ circle, and we divide the circumference (the distance around) by
the diameter (the distance across), the result is _always_ pi:

  circumference   distance around
  ------------- = --------------- = pi
    diameter      distance across 

Note that for this to work out, we have to measure both distances in
the same units!  (For example, if we have the circumference in feet,
and the diameter in inches, the ratio will not be equal to pi.)

The important thing to understand is that ANY circle, no matter how
big or small, always gives the same answer when you divide its
circumference by its diameter.  As the circle gets bigger or smaller,
the circumference and diameter both increase or decrease together, and
their ratio stays the same.  This constant ratio or division answer is
what we call "pi".

There are several approximations for the actual value of pi that are
commonly used.  One of them is 22/7, others are 3.14 or 3.14159. 
Remember that all of these are just approximations, as the actual
value of pi has an infinite decimal expansion that does not go into a 
repeating pattern.  Pi starts out 3.14159... so that's a good approximation 
but still not completely accurate.  3.14 is less accurate, and 22/7, which
when divided gives 3.142857... is also less accurate.  

Since pi always has the same value, if we know either the
circumference or the diameter of a circle we can find the other, by
writing the above equation with the values we know, and solving for
the one we don't.  

For example, suppose we know that the diameter of a circle is 12 cm,
and we want to know the circumference:

       circumference
  pi = ------------- 
          12 cm

If we multiply both sides of this equation by 12 cm, the denominator
on the right cancels out:

               circumference
  pi * 12 cm = ------------- * 12 cm 
                  12 cm

               circumference * 12 cm
  pi * 12 cm = --------------------- 
                        12 cm

                               12 cm
  pi * 12 cm = circumference * ----- 
                               12 cm

                               
  pi * 12 cm = circumference * 1 


  pi * 12 cm = circumference 

Now, we can just leave it like this, or we can use an approximate
value for pi to get an approximate value for the circumference, e.g., 

  3.14 * 12 cm = 37.68 cm
                               
Okay, so what if we know the circumference instead, and we want to get
the diameter?  For example, suppose we know that the circumference is
42 cm.  Then we have

         42 cm
  pi = --------
       diameter

Doing the same sort of thing as before, we can multiply both sides by
the diameter, to get

  pi * diameter = 42 cm

and then divide both sides by pi to get

             42 cm
  diameter = ----- = (42/pi) cm
              pi

Again, we could go ahead and use an approximate value for pi,


  diameter = (42 / 3.14) cm

           = 13.38 cm

Does this make sense so far?  The relationship between diameter and
radius is much simpler.  Recall that the diameter is just twice the
radius, which means that the radius is half the diameter.  So to get
one from the other, we just multiply or divide by 2. 

Why do we care about the radius?  Well, to find the area of a circle,
we need to use a second formula:

  area = pi * radius^2

       = pi * radius * radius

So a typical problem might be:

  The circumference of a circle is 42 cm.  What is the area
  of the circle?

To solve this, we would use the circumference to get the diameter, as
we did above:  13.38 cm.  Then we would divide it by two to get the
radius: 6.69 cm.  Then we would plug that into the formula for area:

  area = pi * 6.69^2

       = 3.14 * 140.5 cm^2

Were you able to follow that?  (Let me know if any of this has been
unclear.)  

This might be a good time to point out that although it's tempting to
just plug in an approximate value for pi, especially when you have a
calculator handy, there are good reasons for avoiding that for as long
as possible.

Let's look again at the problem we just solved.  We're told that the
circumference is 42 cm.  That means the diameter is 42/pi cm.  And the
radius is half that, or 21/pi cm.  So the area is 

  area = pi * (42/pi)^2

              42   42
       = pi * -- * --
              pi   pi

         pi * 42 * 42
       = ------------
         pi * pi


         42 * 42
       = -------            Two of the pi's cancel!
         pi
              
So now we just enter pi once at the end, and we don't have to worry
about errors introduced by using an approximate value in the middle. 

Since pi shows up in lots of different formulas, there's a pretty good
chance that in any problem with multiple steps, a pi that shows up in
one step will get canceled by a pi that shows up in a later step.  So
by just leaving it in there, you can save yourself work, and reduce
the opportunity to introduce errors.  

Just to show how you can go in the other direction, let's try starting
with the area instead of the circumference:
 
   The area of a circle is 100 cm^2.  What is the circumference
   of the circle?

Now we start with the same formula,

     area = pi * radius^2

      100 = pi * radius^2

We divide both sides of the equation by pi to get

   100/pi = radius^2

What now?  We take the square root of both sides:

     ______     ________
   \/100/pi = \/radius^2

        ___
      \/100 
      ----- = radius
        __
      \/pi

        
        10  
      ----- = radius
        __
      \/pi

I hope this helps!  Write back if you have more questions, about this
or anything else. 

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