Using a tensor virial method of moments, an approximate solution to the relaxed elastic state of embedded ellipsoidal inclusions is presented that incorporates surface∕interface energies. The latter effects come into prominence at inclusion sizes in the nanometer range. Unlike the classical elastic case, the new results for ellipsoidal inclusions incorporating surface∕interface tension are size-dependent and thus, at least partially, account for the size-effects in the elastic state of nano-inclusions. For the pure dilatation case, exceptionally simple expressions are derived. The present work is a generalization of a previous research that addresses simplified spherical inclusions. As an example, the present work allows us, in a straightforward closed-form manner, the study of effect of shape on the size-dependent strain state of an embedded quantum dot.