This study focusses on characterising numerically the attractor volume in the state space of dynamical systems excited by additive white noise. A definition for stochastic attractors is introduced in terms of probability measure and numerical methodologies are presented to characterise them. The study is limited to investigating the effects of additive noise on fixed point and limit cycle attractors of the corresponding noise-free system. These results are useful in quantifying the probability of hopping of trajectories between different attractors - a phenomenon that is widely observed in stochastically excited dynamical systems.