Cross-over trials in clinical medicine, have gained interest amongst researchers because of their advantages over parallel group studies, in suitable scenarios. Although many authors have discussed crossover studies with three or more treatments, there has been a lack of comprehensive discussion about sample size calculation for such designs. This thesis proposes a method to calculate the most optimal sample size for a Three-Treatment Three-Period Cross-over Design with Normal Response, on the hypothesis to detect differences between the treatments. A comparison with Parallel Group Design is also presented to show how the corresponding cross-over design requires fewer subjects to achieve the same power. Finally, the robustness of the proposed sample size calculation methods were tested on thousands of simulated samples and different design scenarios, using SAS®. For future research, similar approaches could be followed for discrete responses and/or with three or more than three treatments.