In Stone Circle Design and Measurement, G J Bath says

The Ancient Egyptians developed a procedure to determine the area of a circle by subtracting one-ninth from the diameter and squaring the result. 

This can be best visualised as:

CircleFromSquare equalarea

Figure 1 The near integer relationship between the half side length of a square and the circle of equal area.

This shows that a right angled triangle with longer sides of eight and nine effectively defines the dimensions of any circle equal in area to a square, and visa versa. However, the two areas are not then quite equal. We will first calculate their inequality and then propose a change of measurement units within the same module, to correct this inequality using ancient metrology. 

The Inequality of Areas

If we ignore the irrational PI and use the simplest most accurate approximation for PI as equal to 22/7 then 

  1. The area of the sub-squares is (8/9), its side length, squared of the circle radius or 64/81, whilst
  2. The area of the sector is 22/7 times (1/2) squared or 11/14.
  3. These fractions do not perfectly cancel to 1 but leave a fraction 64/81 x 14/11 = 896/891 = 1.00561167 and
  4. We notice the fractional part is 179/178 which is very close to the 176/175 microvariation used in ancient metrology and the model of the Earth.

If one divides 1.00561167 by 176/175 (= 1.00571428) one obtains 9800/9801 which is unity to one part in 9800, nearly one part in ten thousand and hence virtually exact.

Therefore, from the point of view of ancient numeracy and metrology, the square has 256 square units whilst the circle of 9 units has an area of 22/7 x 92 (81) = 254.571428 which is 175/176 less than 256. That is, the area of the circle is less by 891/896 = 0.9944196. However, if one increaded the units of the radius 9 by 176/175 (using the root canonical) the result, being squared, overshoots beyond 257. But using 441/440, the standard value, the area of the circle is 255.73, just one part in a thousand wrong.and given that 22/7 is larger than the actual Pi, the area would be 255.83. 

Its Correction

  • To obtain a fair equal area between a circle radius 9 and a square half side length 8, one should measure that 8 in root feet and the radius 9 in standard feet, 441/440 the greater. 

The value of this geometrical knowledge is only symbolic since metrology has no difficulty forming a right triangle is longer sides equal to 8 and 9 by forming that triangle. Visually though, one can see in figure 1 that the 9 side, in getting larger is pushing upwards the intersection between the circle and the square and the circle only is getting larger in area.

Finally, this ratio is very important to music as the Pythagorean tone 9/8 but also astronomically in giving the ratio of day counting between the lunar year and Jupiter.