We study the Service Rate Region of Reed-Muller codes in the context of distributed storage systems. The service rate region is a convex polytope comprising all achievable data access request rates under a given coding scheme. It represents a critical metric for evaluating system efficiency and scalability. Using the geometric properties of Reed-Muller codes, we characterize recovery sets for data objects, including their existence, uniqueness, and enumeration. This analysis reveals a connection between recovery sets and minimum-weight codewords in the dual Reed-Muller code, providing a framework for identifying those recovery sets. Leveraging these results, we derive explicit and tight bounds on the maximal achievable demand for individual data objects, thereby defining the maximal simplex within the service rate region and the smallest simplex containing it. These two provide a tight approximation of the service rate region of Reed-Muller codes.