We report on the enhanced light extraction of GaN-based light emitting diodes (LEDs) fabricated with increased mesa sidewalls. It is shown that light extraction and operating voltages in LEDs typically have an inverse relationship that depends on the perimeter of the mesa sidewalls and the mesa area. Based on these observations, we designed an advanced LED incorporating regularly arrayed circular mesa holes to simultaneously maximize the mesa sidewall perimeter and area. Compared to a reference LED, the output power of our mesa-hole LED was enhanced by 12%, and its operating voltage was decreased by 0.1 V, resulting in a power efficiency improvement of 8.8%.