  We derive an exact formula for the corner free-energy contribution of
weakly-anisotropic two-dimensional critical systems in the Ising universality
class on rectangular domains, expressed in terms of quantities that specify the
anisotropic fluctuations. The resulting expression agrees with numerical exact
calculations that we perform for the anisotropic triangular Ising model and
quantifies the nonuniversality of the corner term for anisotropic critical
two-dimensional systems. Our generic formula is expected to apply also to other
weakly-anisotropic critical two-dimensional systems that allow for a conformal
field theory description in the isotropic limit. We consider the three-states
and four-states Potts models as further specific examples.
