First formulated in 1924, the Banach-Tarski paradox states that it is possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five by Robinson (1947), although the pieces are extremely complicated. (Five pieces are minimal, although four pieces are sufficient as long as the single point at the center is neglected.) A generalization of this theorem is that any two bodies in that do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other.
Of course, this doesn't actually work for everyday physical objects, because they are built up out of atoms, not an infinite continuum of points, but it does work for the idealized objects you are accustomed to studying in high-school geometry class!