The energetics of thin elongated ferromagnetic nano-islands is considered for some different shapes, aspect ratios and applied magnetic field directions. These nano-island particles are important for artificial spin ice materials. For low temperature, the magnetic internal energy of an individual particle is evaluated numerically as a function of the direction of a particle's net magnetization. This leads to estimations of effective anisotropy constants for (1) the easy axis along the particle's long direction, and (2) the hard axis along the particle's thin direction. A spin relaxation algorithm together with fast Fourier transform for the demagnetization field is used to solve the micromagnetics problem for a thin system. The magnetic hysteresis is also found. The results indicate some possibilities for controlling the equilibrium and dynamics in spin ice materials by using different island geometries.