But our main focus is the royal foot (remen) whose use can be seen in the design of the pyramids of the Old Kingdom. Let's take a look at the Great Pyramid of Giza whose original dimensions are estimated to have a base of 230.5 m and a height of 146.6 m. What does this tell us? Well, the ratio of the base to the height is 1.57231 which doesn't say much by itself but we can look at its continued fraction.
which shows that one gets a best fit for 11/7 with a scale factor of 40 cubits or 70 feet. This implies that in Khufu's time,
1 royal foot = 299.35 mm
1 royal cubit = 523.86 mm
which is very close to the values for the cubit stick from the 18th Dynasty.
An interesting coincidence is that,
11/7 = 1.5714
π/2 = 1.5708