The influence of the anisotropy in the equilibrium and stability of strange stars is investigated through the numerical solution of the hydrostatic equilibrium equation and the radial oscillation equation, both modified from their original version to include this effect. The strange matter inside the quark stars is described by the MIT bag model equation of state. For the anisotropy two different kinds of local anisotropic σ = pt - pr are considered, where pt and pr are respectively the tangential and the radial pressure: one that is null at the star's surface defined by pr(R) = 0, and one that is nonnull at the surface, namely, σs = 0 and σs ≠ 0. In the case σs = 0, the maximum mass value and the zero frequency of oscillation are found at the same central energy density, indicating that the maximum mass marks the onset of the instability. For the case σs ≠ 0, we show that the maximum mass point and the zero frequency of oscillation coincide in the same central energy density value only in a sequence of equilibrium configurations with the same value of as. Thus, the stability star regions are determined always by the condition dM/dpc > 0 only when the tangential pressure is maintained fixed at the star surface's pt(R). These results are also quite important to analyze the stability of other anisotropic compact objects such as neutron stars, boson stars and gravastars.