Wireless sensor networks (WSNs) are becoming increasingly utilized in applications that require remote collection of data on environmental conditions. In particular dense WSNs are emerging as an important sensing platforms for the Internet of Things (IoT). WSNs are able to generate huge volumes of raw data, which require network structuring and efficient collaboration between nodes to ensure efficient transmission. In order to reduce the amount of data carried in the network, data aggregation is used in WSNs to define a policy of data fusion and compression. In this paper, we investigate a model for data aggregation in a dense {WSN} with a single sink. The model divides a circular coverage region centered at the sink into patches which are intersections of sectors of concentric rings, and data in each patch is aggregated at a single node before transmission. Nodes only communicate with other nodes in the same sector. Based on these assumptions, we formulate a linear programming problem to maximize system lifetime by minimizing the maximum proportionate energy consumption over all nodes. Under a wide variety of conditions, the optimal solution employs two transmissions mechanisms: direct transmission, in which nodes send information directly to the sink; and stepwise transmission, in which nodes transmit information to adjacent nodes. An exact formula is given for the proportionate energy consumption rate of the network. Asymptotic forms of this exact solution are also derived, and are verified to agree with the linear programming solution. We investigate three strategies for improving system lifetime: nonuniform energy and information density; iterated compression; and modifications of rings. We conclude that iterated compression has the biggest effect in increasing system lifetime.