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Cavalieri's principle will tell you that the shape of the base doesn't matter. The area of the base is the only information about the base that is needed.

Cavalieri's principle (which I suspect was enunciated by Archimedes of Syracuse a couple of millennia before the man named Cavalieri was born) says that if every horizontal cross-section of one three-dimensional figure has the same area as the same horizontal cross-section of another three dimensional figure, then the two figures have the same volume.

Suppose you have two pyramids whose bases have equal areas (but do not necessarily both have the same shape), and that have equal heights. Now suppose, for example, that you look at a horizontal plane ("horizontal" means parallel to the base) that is $1/3$ of the way from the apex down to the base. The intersection of the pyramid with that plane has the same shape as the base, but every distance between two points in that intersection is $1/3$ times the distance between the corresponding pair of points in the base. Consequently the area of that intersection is $1/9$ the area of the base. That applies to both pyramids: the areas of the intersections are the same in both. Thus if one of them is one-third-base-times-height, then so is the other.

Thus in order to prove, without using antiderivatives, that the volume is $Ah/3,$ it is enough to prove it with just one pyramid, with just one shape of the base. That can be done as follows: the cube $0\le x\le1,\,\,0\le y\le1, \,\,0\le z\le1$ can be partitioned into three congruent pyramids, as follows: The base of one of them is the unit square in the $(x,y)$-plane and the apex is $(x,y,z)=(1,1,1).$ The base of another is the unit square in the $(x,z)$-plane and the apex is that same point, $(x,y,z)=(1,1,1).$ The base of the third is the unit square in the $(y,z)$-plane and the apex is again that same point. Thus each has one-third of the volume of the whole cube.