In the archaeological community there is an interest in knowing the volume
of the vessels rescued in the excavations, to study, among other things, if
the capacity measures were standardized. The problem is that sometimes
it is not possible to check the volume physically because the piece is too
delicate or too large or simply incomplete. There is a fairly widespread
idea that it is sufficient to know a radial section of the vessel to reconstruct it in 3D. Then, the volume is approximated by dividing the height
in small sub-intervals sufficiently small and in each sub-interval the volume
is approximated by that of a cylinder. This is equivalent to using the composite rectangular rule for the squared radius. This method works well if the
piece made around is regular, more specifically if the horizontal sections are
circumferences. The point is that most archaeological pieces have deformations, which makes the previous method very inaccurate. In this work a
method that, instead of using a single section, uses several radial sections
equally spaced between 0 and 2π angles and then take the average, is proposed. It is shown that the method gives a volume approach of fourth order
with respect to the angle. Numerical experiments are presented on an academic example whose horizontal sections are ellipses, another academic
example with less symmetries and an example of a Pintia vessel with an
evident deformation in which the proposed method is tested.
