Since wireless sensor networks consist of a large number of tiny sensors with limited power supply, it becomes a major concern that how to extend sensor network lifetime and maintain sufficient sensing area at the same time. To achieve this goal, a broadly used strategy is to select some sensor nodes as working nodes to cover the deployment area and at the same time turn off redundant nodes. This paper proposes a mathematical model in which only if the proportion of the sensing range of nodes to the range of the deployment area is known, the number of the nodes needed to reach the expected coverage fraction can be calculated. This work is different from the most literature studying the coverage problem because it is not based on location information of sensor nodes, and thus the cost of hardware and the energy consumption on sensor nodes for deriving and maintaining location information can be considerably reduced. The simulated experiment suggests that in random deployment strategy, the error between the expected coverage fraction and the coverage fraction derived from the simulated experiment is no larger than 2% of the expected coverage fraction; when the expected coverage fraction and the coverage fraction derived from the simulated experiment is the same, the error between the number of working nodes derived from the calculation and the number of working nodes derived from the simulated experiment is less than 5% of the number derived from the calculation. It suggests that the results are identical to the experimental results. The analytical results in this paper can be widely adopted in handling sensor deployment, topology control, and other issues.