Egyptian Equivalent of Pi

Date: 05/15/2003 at 22:09:15
From: Willabee
Subject: Pi 

The ancient Egyptians used the formula (d-d/9)^2 for the area of a
circle with diameter d. What value for pi does the formula yield?

Date: 05/16/2003 at 12:54:43
From: Doctor Dotty
Subject: Re: Pi 

Hi Willabee,

Thanks for the question,

We currently use the formula:

   a = Pi * r^2

Where r is the radius, and a is the area of the circle.

We are told the Ancient Egyptians used:

   a = (d - d/9)^2

Let's multiply out the brackets:

   a = (d - d/9)(d - d/9)

   a = (d)(d) - (d/9)(d) - (d/9)(d) + (d/9)(d/9)

             d^2   d^2   d^2
   a = d^2 - --- - --- + ---
              9     9     81

             2d^2   d^2
   a = d^2 - ---- + ---
               9     81

Now let's look again at our present formula:

   a = Pi * r^2

It is in terms of the radius (r), and the Egyptian one is in terms 
of the diameter (d). We know that 2r = d (write back if you would 
like this explained), so we can rewrite the Egyptian equation as:

                2(2r)^2   (2r)^2
   a = (2r)^2 - ------- + ------
                   9        81

              8r^2   4r^2
   a = 4r^2 - ---- + ----
               9      81

Common denominate:

       4r^2   8r^2   4r^2
   a = ---- - ---- + ----
        1      9      81

       324r^2   72r^2   4r^2
   a = ------ - ----- + ----
         81       81     81

Can you collect the terms, and find what the Egyptian equivalent of Pi 
is from here?

Write back if I can be of any more help - on this or anything else.

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