[en] Highlights: • The temperature-versus-misfit strain phase diagrams of an epitaxial (Ba_0.50Sr_0.50)TiO₃ dot were constructed. • Polarization and toroidal moment are sensitive to boundary conditions, resulting in a wide variety of phases. • Within a narrow range of surface charge screening an intermediate phase in which polarization and toroidal moment coexist. • Effect of misfit strain and electrical boundary conditions on ferroelectric properties at room temperature are investigated. A first-principles-derived effective Hamiltonian approach is used to reveal the temperature-versus-misfit strain phase diagram of an epitaxial (Ba_0.50Sr_0.50)TiO₃ dot under different electrical boundary conditions. The results indicate that the electrical polarization and toroidal moment are highly sensitive to the applied strain and/or electrical boundary conditions, resulting in a wide variety of phases that are not found in a free-standing BST dot and in bulk. The calculations indicate that within a narrow range of surface charge screening an intermediate phase in which the polarization and toroidal moment coexist. The dependences of the electrical polarization, toroidal moment and dielectric permittivity on the misfit strain and electrical boundary conditions at room temperature are also investigated and compared with the available theoretical predictions and experimental measurements.