Base station placement has significant impact on sensor network performance. Despite its significance, results on this problem remain limited, particularly theoretical results that can provide performance guarantee. This paper proposes a set of procedure to design (1 - ) approximation algorithms for base station placement problems under any desired small error bound > 0. It offers a general framework to transform infinite search space to a finite-element search space with performance guarantee. We apply this procedure to solve two practical problems. In the first problem where the objective is to maximize network lifetime, an approximation algorithm designed through this procedure offers 1 / 2 complexity reduction when compared to a state-of-the-art algorithm. This represents the best known result to this problem. In the second problem, we apply the design procedure to address base station placement problem for maximizing network capacity. Our (1 - ) approximation algorithm is the first theoretical result on this problem.