Mobile chargers (MCs) are usually dispatched to deliver energy to sensors in wireless rechargeable sensor networks (WRSNs) due to its flexibility and easy maintenance. This paper concerns the fundamental issue of charging path DEsign with the Minimized energy cOst (DEMO), i.e., given a set of rechargeable sensors, we appropriately design the MC’s charging path to minimize the energy cost which is due to the wireless charging and the MC’s movement, such that the different charging demand of each sensor is satisfied. Solving DEMO is NP-hard and involves handling the tradeoff between the charging efficiency and the moving cost. To address DEMO, we first investigate how to identify a single charging position where the MC could stay to charge a set of sensors distributed within a small area with the maximized charging efficiency. Then, based on the result obtained in the case of optimizing a single charging position, we develop a computational geometry-based algorithm to deploy multiple charging positions within the whole network, by considering the fixed and finite charging range of the MC. We prove that the designed algorithm has the approximation ratio of $O\!\left ({\ln \!N}\right)$ , where $N$ is the number of sensors. Then we construct the charging path by calculating the shortest Hamiltonian cycle passing through all the deployed charging positions within the network. In addition, we investigate the impact of the network topology as well as the distribution of charging demands among sensors on the MC’s energy cost during a charging tour. Extensive evaluations validate the superiority of our path design in terms of the MC’s energy cost minimization, compared with existing main algorithms.