The coffer's dimensions represent 2 ideas 1. pi and 2. the Radius of the Earth.
In another post some time ago I noted that a sphere with a radius of the exact value for pi in MILES 3141.592653 miles gave a volume which is half that of the planet Earth.
The volume of a sphere with radius 3141.592653 = 129,878,788 cubic miles and double that = 259,757,576 cubic miles the volume of the Earth with a radius of 3,958.15874 miles.
The coffer has these numbers Quote'61. From the foregoing data the cubic quantities may be calculated of a simple rectilineal box, omitting all notice of the attachments for the lid, employing the mean planes :–
Contents– 72,030; solid bulk = 70,500; volume over all, 142,530 cubic inches. Or by the caliper results, instead of the mean planes, the bulk is 1/580 more, and the contents probably about 1/1000 less; hence the quantities would be :–
Contents = 71,960; solid bulk = 70,630; volume over all, 142,590.
These quantities have a probable error of only about 60 cubic inches on contents and volume, and 100 inches on the bulk. The bulk of the bottom is = 23,830; and hence one side and end is on an average = 23,335. Bulk of bottom x 3 is then = 71,490; and 3/2 x bulk of sides and ends = 70,000, subject to about 100 cubic inches probable error.'
As you may see Petrie allows an error of 60 cubic inches. This is where the Indus/Saxon foot comes into it's own as
pi 3.141592653 Indus feet x 13.2 inches = 41.46902303 inches cubed = 71,313.44416 cubic inches the average between Petrie's solid bulk and contents 71,960 + 70,630 = 142,590 / 2 = 71,295 well within Petrie's margin of error.
Volume over all 71,313.44416 x 2 = 142626.888 cubic inches again well within the margin of error.
The cube root of 142626.888 = 52.24763503 (note the number you came close to) / 13.2 = 3.958158714 the radius of the earth 3958.158714.
Circumference then 24869.84468 miles 1 degree / 360 = 69.082901 miles and 30 seconds of that 3039.64 imperial feet the socket perimeter side = 9118.943 Petrie gives 9125.9.
It is all down to the exact value for pi which the ancient architect may have used.
Look at the number for an approximate for pi 3.142696805 and take away the 3 then 142,696.805 cubic inches is the volume over all of the coffer