We present expressions for the elastic and first-order piezoelectric tensor in (111)-oriented III-V zinc-blende semiconductors. Moreover, an equation for the second-order piezoelectric polarization vector in these systems is derived. Together these expressions provide an efficient route to calculate built-in potentials and strain fields in (111)-oriented zinc-blende nanostructures. Our detailed analysis provides insight into the key parameters that modify strain and built-in fields in a (111)-oriented zinc-blende system compared to a conventional (001) structure. We show that the calculated strain field in a (111)-oriented quantum dot displays the correct C-3v symmetry of the underlying crystal structure, even though we use a continuum-based approach and the quantum dot geometry is higher in symmetry than C-3v, e.g., C-infinity v. This behavior originates from an in-plane angle dependence of certain elastic tensor components in the (111)-zinc-blende system. In addition, we compare the elastic and the first-order piezoelectric tensor of the (111)-zinc-blende systems with the corresponding quantities in a wurtzite structure and point out similarities and differences. This comparison provides, for example, insight into the sign of the shear piezoelectric coefficient e(15) in the wurtzite system, which is still under debate in the literature. Our analysis indicates e(15) < 0, in accordance with recent experimental and theoretical results.