The Q of an electrically small planar antenna cannot approach the Q bounds as obtained for spherical shapes for the same value of ka. Comparing the Q of planar antennas with this limit does not provide sufficient insight into how well the antenna has been miniaturized relative to the theoretical performance that can be achieved with the planar shape. In this paper, a new bound for radiation O of planar antennas is presented, according to Gustafsson's bounds for arbitrary shapes. In addition, a suitable Figure of Merit for antenna miniaturization is defined and several designs for electrically small, thin planar antennas are presented.