Cavalieri's principle
/ kav-uh-lee-AIR-ee /
Stack a deck of cards into a neat rectangular block, then push the side so the deck leans into a slanted parallelogram-shaped pile. The pile holds exactly the same number of cards — the same volume — because you only slid the cards sideways, you did not add or remove any. Cavalieri's principle is the precise statement of this everyday fact.
It says: if two solids sit between the same pair of parallel planes, and every plane parallel to those two cuts both solids in cross-sections of equal area, then the two solids have equal volume. (The two-dimensional version: two regions of equal height whose every horizontal slice has equal width have equal area.) The slices need not have the same shape — only the same area at each level. This lets you compare a solid you understand with one you do not.
It is the workhorse behind volume formulas that pre-date calculus. The volume of an oblique prism equals that of an upright one (slide the slices straight). The (4/3)·pi·r^3 of a sphere falls out by matching a hemisphere, slice for slice, against a cylinder with a cone carved out. Cavalieri reasoned with 'indivisibles' in the 1600s; the idea is sound, though making 'equal cross-sections at every level' fully rigorous is the business of the integral, which is why the airtight proofs belong to analysis.
Two stacks of identical coins, one stacked straight and one leaning, occupy the same volume: at every height each stack shows one coin of identical cross-section, so by Cavalieri their volumes match.
Equal cross-sections at every level mean equal volume — shape need not match.
It compares solids of equal height with matching cross-sectional areas at every level; if the heights differ or some level's areas differ, the conclusion fails — it is about areas at each slice, not about overall shape.