In this paper, we shall introduce a new geometric constant \(A_{2}(X,B)\) , which is closely associated with the modulus of smoothness \(\rho _{X}(t,B)\) related to Birkhoff orthogonality, and investigate it in relation with the skewness s(X) by Fitzpatrick and Reznick, the parameter \(A_{2}(X)\) by Baronti et al. and other existing constants. We quantify the characterization of uniform non-squareness in terms of this new coefficient. In the meantime, some precise values of the parameter are computed for X being some specific spaces. Meanwhile, we discuss some applications of \(A_{2}(X,B)\) . In the end, we aslo consider the parameter \(\beta _{X}(B)\) , which is closely connected with the \(\beta _{X}\) modulus of convexity related to Birkhoff orthogonality.
