A polymeric microneedle has been developed for drug delivery applications. The ultimate goal of the polymeric microneedle is insertion into the specified region without failure for effective transdermal drug delivery. Mechanical failure of various geometries of microneedles by axial load was modeled using the Euler formula and the Johnson formula to predict the failure force of tapered-column microneedles. These formulas were compared with measured data to identify the mechanical behavior of microneedles by determining the critical factors including the actual length and end-fixed factor. The comparison of the two formulas with the data showed good agreement at the end-fixity (K) of 0.7. This value means that a microneedle column has one fixed end and one pinned end, and that part of the microneedle was overloaded by axial load. When the aspect ratio of length to equivalent diameter is 12:1 at 3 GPa of Young's modulus, there is a transition from the Euler region to the Johnson region by the decreased length and increased base diameter of the microneedle. A polymer having less than 3 GPa of stiffness would follow the Euler formula. A 12:1 aspect ratio of length to equivalent diameter of the microneedle was the mechanical indicator determining the failure mode between elastic buckling and inelastic buckling at less than 3 GPa of Young's modulus of polymer. Microneedles with below a 12:1 aspect ratio of length-to-equivalent diameter and more than 3 GPa of Young's were recommended for reducing sudden failure by buckling and for successfully inserting the microneedle into the skin.