Petrie set himself the task of proving or disproving Smyth's pi theory in which the shape of the Great Pyramid, the design of the King's Chamber, and the volume of the Coffer, were all said to be based on pi = 3.14159..
Petrie concluded that the pi ratio evident from the design was 22/7. The long walls of the King's Chamber have a length of 560 digits (20 royal cubits) and a perimeter of 1,760 digits in the pi ratio 22/7. (The height of the walls is 320 digits, made up of 5 very precise courses of granite blocks of 64 digits.) This mirrors the shape of the Great Pyramid on a scale of 1 digit to 1 royal cubit. The perimeter of the base of the pyramid is 1,760 royal cubits (4 x 440) which was interpreted as the circumference of a circle, and 1,760 divided by 2pi is 280 as the height of the pyramid, interpreted as the radius of the circle.
Petrie noted that the shape of the Coffer strongly bears out the pi theory. The internal area is equal to the area of a circle with a diameter of 2 1/2 royal cubits, and the volume is equal to a sphere with a diameter of 2 1/2 royal cubits - 'by the true contents this would need a cubit of 20.644, which is very close to the best determinations'
The internal diagonal is very close indeed to 4 royal cubits (112 digits). Some years ago I noticed that 112 x 2 1/2 = 280 as if the diagonal was a symbolic mulitiplier. It follows that a circle with a diameter of 280 royal cubits may have been seen as equal to the triangular cross-section of the Great Pyramid by using the pi ratio 22/7.
Smyth attributed the discovery of the proportions of the Coffer to the seventeenth century survey of Greaves. Smyth spotted the pi proportions of the King's Chamber, as far as I know.
The three professors named above had very different views about the Great Pyramid, but all measured the Coffer and came to the same conclusion about its shape. I note that it would have been possible to hollow out a hemisphere and measure its volume by filling with sand. Alternatively, as we know that AE could calculate the volume of a pyramid, and even had a formula for the volume of a truncated pyramid, then it may be that the volume of a sphere was considered as many needle shaped pyramids, rather like obelisks. ie the volume of a sphere is the same as a pyramid in which the base of the pyramid is the surface area of the sphere and the height of the pyramid is the radius of a sphere.
Petrie noted that objects such as diorite bowls showed clear evidence of having been worked on a lathe, and yet no such device has been found. This did not stop Petrie from concluding that a lathe had been used. The Coffer was made from a single piece of granite - using a huge tubular drill according to Petrie. The technology of huge drills and precision lathes has not been substantiated by the discovery of the tools, but it would be ridiculous to argue that AE did not have such tools. Petrie did not assume that the Greeks were the first to discover pi, the volume of sphere, etc. He simply looked at the evidence and weighed it in the balance.
'Weighed in the balance and found wanting' is a popular phrase from the judgement of Belshazzar king of Babylon recorded in the Bible. Ideas flowed out of Egypt with the Israelites and others, including the idea of a soul of a person being weighed in the balance. Likewise, mathematical ideas may have come out of Egypt, or may have been discovered independently. There is no proof that the Babylonians were the first to divide the circle into 360 degrees. The idea may have come from Egypt, or may have been decided on independently. My hypothesis on the shape of the Bent Pyramid is based on dimensions in digits, and does not rely on pi ratio 22/7. Is it not true that the peoples of all nations had the ability to discover elementary geometry? Why should anyone argue that the Greeks and the Babylonians thought so very differently from the Ancient Egyptians?