The geometries, stabilities, and electronic and magnetic properties of europium encapsulated EuSi(n) (n=1-13) clusters have been investigated systematically by using relativistic density functional theory with generalized gradient approximation. Starting from n=12, the Eu atom completely falls into the center of the Si frame, i.e., EuSi(12) is the smallest fully endohedral Eu silicon cluster. The interesting finding is in good agreement with the recent experimental results on the photoelectron spectroscopy of the europium silicon clusters [A. Grubisic, H. P. Wang, Y. J. Ko, and K. H. Bowen, J. Chem. Phys. 129, 054302 (2008)]. The magnetic moments of the EuSi(n) (n=1-13) clusters are also studied, and the results show that the total magnetic moments of the EuSi(n) clusters and the magnetic moments on Eu do not quench when the Eu is encapsulated in the Si outer frame cage. It is concluded that most of the 4f electrons of the Eu atom in the EuSi(12) cluster do not interact with the silicon cage and the total magnetic moments are overwhelming majority contributed by the 4f electrons of the Eu atom. According to the binding energy per atom, the second difference in energy (Delta(2)E), and vertical ionization potential, the EuSi(n) (n=4,9,12) clusters are very stable.