This study presents a set of closed-form exact solutions for pricing discretely sampled variance swaps and volatility swaps, based on the Heston stochastic volatility model with regime switching. In comparison with all the previous studies in the literature, this research, which obtains closed-form exact solutions for variance and volatility swaps with discrete sampling times, serves several purposes: (1) it verifies the degree of validity of Elliott et al. (2007)'s continuous-sampling-time approximation for variance and volatility swaps of relatively short sampling period; (2) it examines the effect of ignoring regime switching on pricing variance and volatility swaps; (3) it contributes to bridging the gap between Zhu and Lian (2010)'s approach and Elliott et al. (2007)'s framework; (4) and finally it presents a semi-Monte-Carlo simulation for the pricing of other important realized variance based derivatives.