Hi Kanga,

Thanks for your post.

Yes, the formula for volume of a pyramid is apparent from the special case of the division of a cube into 6 pyramids with a slope of 45 degrees (seked of 7 palms).

Empirical evaluations may have revealed that the volume of all pyramids can be calculated from this formula.

The volume of a pyramid is mean cross-sectional area multiplied by height so it follows that all square based true pyramids have the same mean cross-sectional area irrespective of slope which is:
(side length of base) x (side length of base) x 1/3

If a series of pyramids with a base side length of say 10 have heights of say 5, 10, 15 and so on then the plan view of all the pyramids is the same irrespective of slope as apparent from the division of the vertical height into say 8 equal divisions and drawing the horizontal plane for each of those divisions. This explains why the mean cross-sectional area is the same irrespective of slope, so the formula for the special case of 45 degrees applies to all pyramids.

What did Gillings conclude with regard to the method of calculation of the volume of a truncated pyramid in the Moscow Mathematical Papyrus?

Mark