A standard intermediate step in component reliability analysis involves constructing a first-order or second-order approximation to the limit state surface at one design point. This paper looks into the possibilities for building a several-point, piecewise linear approximation to the limit state surface with the aim of obtaining a better estimate of the probability of failure of the component. The piecewise linear approximation is built at the points of intersection between the limit state surface and the coordinate axes and the one closest to the origin in the standard normal space. Details of finding these points are mentioned and computational results on several problems reported. The experiments suggest the method to be a promising one if care is exercised in applying it, and situations where one needs to be careful are described.