Service rate is an important, recently introduced, performance metric associated with distributed coded storage systems. Among other interpretations, it measures the number of users that can be simultaneously served by the system. We introduce a geometric approach to address this problem. One of the most significant advantages of this approach over the existing ones is that it allows one to derive bounds on the service rate of a code without explicitly knowing the list of all possible recovery sets. To illustrate the power of our geometric approach, we derive upper bounds on the service rates of the first order Reed-Muller codes and the simplex codes. Then, we show how these upper bounds can be achieved. Furthermore, utilizing the proposed geometric technique, we show that given the service rate region of a code, a lower bound on the minimum distance of the code can be obtained.