Solid state lithium ion batteries have high potential for future energy storage since they promise high energy density and safety, latter being related to the mechanical properties of the materials. For example the micro-battery, being one application of the solid state batteries, requires a thin solid electrolyte, which renders especially anisotropic behavior of the materials’ properties important. In particular, Li1+xAlxTi2-x (PO4)3 (LATP) is a promising solid electrolyte material that possesses a rhombohedral crystal structure that might lead to such anisotropic mechanical properties. In current work a nanoindentation test is combined with EBSD technique to correlate elastic modulus and hardness of LATP to the crystal orientation. Furthermore, calculations based on the Vlassak-Nix and easy-slip models are adopted to verify the anisotropic mechanical properties. Overall, the experimentally derived elastic modulus and hardness of LATP show similar trends. The experimentally derived indentation modulus agrees well with the prediction of the Vlassak-Nix model. The normalization of the experimentally derived hardness and the hardness ratio from the easy-slip model reveal comparable trends. This work aids a deepening of the understanding of the mechanical properties of LATP electrolyte and provides a basis for further improvement of the Vlassak-Nix and easy-slip models in the application to rhombohedral materials.