C.B. not quite. One starts with Pi 3.14159265359 first to figure out the obvious pattern.
4/pi = 1.27324 which resembles closely the square root of the Golden Ratio 1.27202
From there, it's really just a question of replacing pi with the result found in 4/sqrt(phi) and test using the perimeter of the square as a basis for the circumference of the Circle and use the circumference (which truly is the same as a square's perimeter) and divide it with 4/sqrt(phi) to find 3.144606, because if we were to take the circumference (which is the same as the perimeter) and divide it by 4/1.27324 (1.27324 being the result found in 4/pi), then the result will eventually be 3.14159265359, however, errors in accuracy will also creep up more often with 3.14159265359 than with 3.14460551103 for the utmost certainty.