In order to study the relation between the core and corona in galactic star clusters, the spatial structure of 38 rich open star clusters has been studied using radial density profiles derived from the photometric data of the Digital Sky Survey. The shape of the radial density profile indicates that the corona, most probably, is the outer region around the cluster. It can exist from the very beginning of the cluster formation and dynamical evolution is not the reason for its occurrence. The study does not find any relation between cluster size and age but indicates that the clusters with galacto-centric distances >9.5 kpc have larger sizes. Further, we find that the average value of the core radius is 1.3+/- 0.7 pc and that of annular width of the corona is 5.6+/- 1.9 pc, while average values of densities of cluster members in the core and corona are 15.4+/- 9.9 star/pc2 and 1.6+/- 0.99 star/pc2 respectively. Average field star contaminations in the core and corona are ~ 35% and 80% respectively. In spite of smaller densities in the coronal region, it contains ~ 75% of the cluster members due to its larger area in comparison to the core region. This clearly demonstrates the importance of the coronal region in studies dealing with the entire stellar contents of open star clusters as well as their dynamical evolution. In contrast to the cluster cores, the structure of coronal regions differs significantly from one cluster to other.