  In this paper we study spherically symmetric metrics on a symmetric space in
$\mathbb{R}^n$ with scalar and constant flag curvature and we also obtain
families of this type of metrics. Many explicit examples are provided for
Douglas metrics with scalar and constant flag curvature. Furthermore, new
examples of projectively flat Finsler metrics are given. We also provide a
family of spherically symmetric Finsler metrics which are not of Douglas type.
