See Hilbert's third problem for what we want to prove, motivation, discussion and importance. Due to the moderate length of the proof, I give an outline of the ideas and principal constructs used. More details are in the linked nodes.

Max Dehn proved a counterexample to Hilbert's third problem; in fact, his method gives a means to construct very many counterexamples, as well as a more general framework for studying how polyhedra can fit together in three dimensions. The proof shows how to construct pairs of tetrahedra of equal volume which are neither equidecomposable nor equicomplementable. As such, the equality of their volumes cannot be justified without the use of calculus.

The proof uses some linear algebra in an ingenious way. It may help to "recall" that the real numbers R are a vector space over the field of rational numbers Q. A basis for this vector space is often called a Hamel basis, to emphasise that only finite linear combinations are considered here (unlike, e.g., the bases used in Hilbert spaces and other Banach spaces). The existence of such a Hamel basis requires the axiom of choice; luckily, Dehn doesn't really require such a basis. The proof can proceed without relying on the axiom of choice. It is completely constructive: the tetrahedra may be specified explicitly.