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### 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/
```
Associated Topics:
Middle School History/Biography
Middle School Pi

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