Photonic crystals are known to enhance the extraction efficiency of LEDs and simultaneously shaping the emission pattern. In order to determine the radiation pattern we developed a model based on coupled mode theory that takes into account the lattice pattern, etch depth and the mode distribution. From a basic geometrical consideration a fundamental limit for the directionality is predicted. The calculations fit well to experimental data obtained from green InGaN LEDs incorporating a hexagonal PhC revealing a maximum directionality of 31% within 30°. Additional FDTD simulations were performed for determining quantitatively the extraction efficiency of PhC LEDs compared to LEDs with a roughened surface. Despite its lower overall extraction efficiency, the PhC LED outperforms a standard LED with surface roughening within an acceptance angle of 34° due to the higher directionality of the extracted light.