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Is Cavalieri's principle (two volumes are equal if every cross section has equal area) considered calculus or not?

I'd say that this is an application of Fubini's theorem which is very clearly calculus (or analysis, not sure where the distinction is drawn in English).

calculus = introductory real analysis aimed at engineering and science students. Same material, different emphasis.

I would say "circular pyramid" is intuitive, but "an n-sided pyramid works just the same as a square pyramid" is not.

If instead of just a square pyramid, you demonstrated it for regular polygons with sides 3, 4, 5 - I wouldn't need any convincing that the relationship continued to infinity.

It's both.