Because of our knowledge of pi related to calculating the area of a circle we can convert the equation to derive an Egyptian Pi value.

Problem 10 in the Moscow papyrus fundamentally resolves to the same Pi value. In the following all references to Pi are approximate:

Egyptian Applied Circle Area Constant (E) = (1-1/9)^2

E = Pi/4 or 4E = Pi

Sqrt(E) = (1-1/9) = 8/9 = sqrt(Pi)/2

What follows here is some speculation:

Sqrt(E) x 2 x 137 x 10^5 Royal Cubits of 20.62 inches is a very good approximation for the diameter of Earth within 100m of the currently accepted radius at the equator multiplied by 2 to achieve diameter.

The currently accepted radius of earth is 6,378,137m according to Wikipedia. Sqrt(E) x 137 x 10^5 RC for radius is 50.2 m out or 0.0007% error.

Through the method of determining area of a circle proposed using a 9x9 grid, we know that the Egyptians knew E and Sqrt (E) without formally naming these values in the Rhind Papyrus. In the above calculation, I have also used 137 which I derived from the height of Khafres Pyramid being 2 x 137 = 274 RC and baselength 411 Royal Cubits = 3 x 137 clearly establishing 137 as an important number.

Before I write this insight off as a coincidence, I have to ask the question of what the value would be of setting a radius/diameter to include a multiple of the square root of Pi?