We report a numerical analysis of the light-extraction efficiency (LEE) of light-emitting diodes (LEDs) on patterned sapphire substrates (PSSs). We considered various n-sided, regular convex pyramids, where n is an integer and $n{geq}3$. We then considered four cross sections: extruded, subtracted, truncated-extruded, and truncated-subtracted. Ray-tracing simulations were carried out with these polygonal pyramid patterns, and the dimensions of the patterns were systematically varied. Optimized pattern shapes were determined for large LEE. An extruded circular pyramid with a slant angle of $45^{circ}$ was found to be the optimal patterned shape.