In Additive Manufacturing (AM), typically a trade-off exists between part quality and build time. Part orientation with respect to the print direction may significantly influence both. In this thesis, the consequences of part orientation on support volume requirements are studied. Build time, material consumption, and post-processing efforts are influenced by the amount and configuration of required supports. Using triangular surface meshes, the support requirement for a given part orientation is calculated for each triangle facet individually and summed. Gradient descent methods are used to optimize part orientation for minimum support volume. To enable implementation of gradient descent optimization, focus is placed on obtaining derivative information of the support volume on a per-facet basis. The resulting support volume function contains discontinuities, for which smooth approximation strategies are implemented. This approach is first applied to convex shapes, with promising results. For non-convex shapes however, non-local information is required. A novel method for indicating the presence of on-part supports is presented. All possible candidates for support on part are computed for each facet before the start of the optimization process. The resulting connectivity set is an inherent property of the shape and only requires calculation once. The new method is tested using numerical experiments, which indicate that gradient-based optimization of the smooth volume function outperforms the population-based approaches commonly used in the literature. Moreover, the presented work provides a framework for optimizing total part cost in which other metrics are easily appended.