Eumelanin is a ubiquitous biological pigment, and the origin of its broadband absorption spectrum has long been a topic of scientific debate. Here, we report a first-principles computational investigation to explain its broadband absorption feature. These computations are complemented by experimental results showing a broadening of the absorption spectra of dopamine solutions upon their oxidation. We consider a variety of eumelanin molecular structures supported by experiments or theoretical studies, and calculate the absorption spectra with proper account of the excitonic couplings based on the Frenkel exciton model. The interplay of geometric order and disorder of eumelanin aggregate structures broadens the absorption spectrum and gives rise to a relative enhancement of absorption intensity at the higher-energy end, proportional to the cube of absorption energy. These findings show that the geometric disorder model is as able as the chemical disorder model, and complements this model, to describe the optical properties of eumelanin.