The classical Steiner formula deals with the volume of $K+rD_n$, where $K$ is a convex compact set in $R^n$, $r\ge0$, and $D_n$ the Euclidean disc in $R^n$. We compute the volume of $(K+rD_n)^*$, where $K^*$ means the dual of $K$. We also represent some Steiner functionals by integrals and prove some inequalities.