In this paper, we calculate the eccentricities and the distance vectors of all vertices of the \( (5,0) \)-nanotubes. Building on these computations, we further determine several important distance-based topological indices associated with these nanotubes. Specifically, we investigate the eccentric connectivity index, eccentric adjacency index, first and second eccentric connectivity indices, Wiener index, generalized Wiener index, generalized Wiener polarity index, hyper-Wiener index, and reciprocal complementary Wiener index. These indices are instrumental in characterizing the structural and connectivity attributes of nanotubes, offering significant insights into their topological properties. The arguments from this paper could be readily adapted to obtain similar results for (6,0)-nanotubes.