...the top half of the pyramid contains, hmm, about 1/9th of the mass, uh, I'd better get someone else to look at that...
Hmmm. I think you may have been fooled by that tricky 1/3 term in the formula for the volume of a pyramid.
V_total = (1/3) x h x B
(B, being the area of the base, h the height).
If we lop off the top half of the pyramid, the resulting solid has half the height, and the area of the base will be one fourth as great, since its two linear dimensions scale proportionally, the volume of the top half of the pyramid is
V_top = (1/3) x (h/2) x (B/4) = V_total/8
This relationship should hold regardless of shape of the base and whether or not the stack ascends vertically, think of a leaning pyramid of Pisa.
For the actual pyramid, the mass may well not be proportional to the volume, since the lower parts of it are honeycombed with passages, the upper part may well represent more than one eigth of the total mass.