This paper introduces and evaluates a fast exact algorithm and a series of faster approximate algorithms for the computation of 3D geometric moments from an unstructured surface mesh of triangles. Being based on the object surface reduces the computational complexity of these algorithms with respect to volumetric grid-based algorithms. In contrast, it can only be applied for the computation of geometric moments of homogeneous objects. This advantage and restriction is shared with other proposed algorithms based on the object boundary. The proposed exact algorithm reduces the computational complexity for computing geometric moments up to order N with respect to previously proposed exact algorithms, from N9 to N6. The approximate series algorithm appears as a power series on the rate between triangle size and object size, which can be truncated at any desired degree. The higher the number and quality of the triangles, the better the approximation. This approximate algorithm reduces the computational complexity to N3. In addition, the paper introduces a fast algorithm for the computation of 3D Zernike moments from the computed geometric moments, with a computational complexity N4, while the previously proposed algorithm is of order N6. The error introduced by the proposed approximate algorithms is evaluated in different shapes and the cost-benefit ratio in terms of error, and computational time is analyzed for different moment orders.