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Surface roughening is frequently employed to increase light extraction from light-emitting diodes (LEDs), especially in the important case of III-Nitride LEDs. We explore the physics governing this scheme. We introduce a numerical model, based on solving Maxwell's equations, to accurately describe scattering by a roughened semiconductor interface. This model reveals the complex angular dependence of the scattering properties. We then couple this approach to an LED light extraction model and predict how surface roughness impacts light extraction. We focus on two important cases, thin-film LEDs and volumetric LEDs. We show that optical losses in the LED dictate light extraction, and that volumetric LEDs offer an opportunity for ultimate efficiency.