In this paper, we shall consider two new constants DW S (X) and DW I (X), which are the Dunkl-Williams constant related to the Singer orthogonality and theisosceles orthogonality, respectively. We discuss the relationships between DW S (X) and some geometric properties of Banach spaces, including uniform non-squareness, uniform convexity. Furthermore, an equivalent form of DW S (X) in the symmetric Minkowski planes is given and used to compute the value of DW S ((R 2 , ∥ · ∥ p)), 1 < p < ∞, and we also give a characterization of the Radon plane with affine regular hexagonal unit sphere in terms of DW S (X). Finally, we establish some estimates for DW I (X) and show that DW I (X) does not necessarily coincide with DW S (X).