The equivalent ellipsoid for magnetized bodies of arbitrary shape can be determined by imposing the equality between the demagnetization factors of the two shapes of equal volume. It is shown that the'commonsense'criterion for mapping two different shapes by imposing the equality of the demagnetization factors for equal aspect ratios often results in large errors. We propose a general method for the rigorous determination of the equivalent ellipsoid. The cases of the exact equivalent ellipsoids for discs, cylinders with elliptical cross section and prisms are worked out and discussed.