We study three different gallium-nitride (GaN) based light emitting diode (LED) cases based on the different locations of the pyramid textures. In case 1, the pyramid texture is located on the sapphire top surface, in case 2, the pyramid texture is locate on the P-GaN top surface, while in case 3, the pyramid texture is located on both the sapphire and PGaN top surfaces. We study the relationship between the light extraction efficiency (LEE) and angle of slant of the pyramid texture. The optimization of total LEE was highest for case 3 among the three cases. Moreover, the seven escape paths along which most of the escaped photon flux propagated were selected in a simulation of the LEDs. The seven escape paths were used to estimate the slant angle for the optimization of LEE and to precisely analyze the photon escape path.