Updated: Sep 27
There are 235 lunar months, or 19 solar years, in a Metonic cycle. This is a cycle of time at the end of which something close to an integer number of lunar months and solar years have elapsed. Cycles such as this form a sort of temporal circular structure, and have been essential in the calculation of calendars historically. Still, you might not expect pi itself, to link the cycles of the sun and the moon, nor perhaps the number of days in a lunar month to become, itself, an multiple of lunar months present in the Metonic cycle.
Yet, to a 99.9999996% accuracy, the number of days in a synodic lunar month, which is 29.53059, if taken as a number of lunar months, i.e. 29.53059 x 29.53059, multiplied by 25 and divided by pi, is the number of days in 19 solar years, i.e. the Metonic cycle. (Using the value for pi given by a calculator). And the number of months in this cycle, which is 235, is very close to 29.53059 x 25 /pi.
29.53059 x 29.53059 x 25 x pi / 365.242199 = 18.99999274
29.53059 x 25 / pi = 234.99696855
This is an almost unbelievable natural coincidence. I found this playing around with numbers on my calculator, on the train home. Made my day!
Curiously, you could interpret the length of one side of the Great Pyramid of Giza, given by Flinders Petrie as 9068.8 inches, as 29.53059 x 29.53059 x 4/3 x 78/10 = 9069.38 inches, which is just over an inch over 29.53059 x 78/10 metres (of the modern kind). In fact, 29.53059 x 4/3 = 39.3812 inches, which is just over a modern metre in inches, of 39.3700787402 inches.