A new modulus of smoothness and its equivalent K-function are defined on the conic domains in \({\mathbb {R}}^d\) , and used to characterize the weighted best approximation by polynomials. Both direct and weak inverse theorems of the characterization are established via the modulus of smoothness. For the conic surface \({\mathbb {V}}_0^{d+1} = \{(x,t): \Vert x\Vert = t\le 1\}\) , the natural weight function is \(t^{-1}(1-t)^{\gamma }\) , which has a singularity at the apex, the rotational part of the modulus of smoothness is defined in terms of the difference operator in Euler angles with an increment \(h/\sqrt{t}\) , akin to the Ditzian-Totik modulus on the interval but with \(\sqrt{t}\) in the denominator, which captures the singularity at the apex.
