Egyptian Circle Mathematics (3 replies)

I have been investigating the Ancient Egyptian calculations of area and volume of shapes that include circles. There is a view that the Ancient Egyptians did not "know" Pi. However, in the Rhind Papyrus problem 50, as has been shown by various mathematicians, a general formula for the Area of a circle is found:

A = (1-1/9)^2 x d^2

I will refer to this presentation which presents a potential method by which the original scribe followed to obtain the result: [www.math.ucdenver.edu]

The approach taken is to look at dividing a square into nine equal squares, then to form an irregular octagon by forming a triangle at each of the corner squares. When I look at this, using simple math, I achieve 7/9 x d ^ 2 as the result which is per the presentation. If you subdivide the corner squares into 9 equal squares again, you end up with 63/9 which the presentation says is approximately equal to 64/9 or (1-1/9)^2. Why did the scribe Ames use (1-1/9)^2 rather than 7/9 if they used the Octagon method proposed?

When I further overlay the (1-1/9) square into the mix, I get the following.



Interesting to note that the square approximately intersects at the points of intersection between the circle and the octagon.