Light scattering by nanostructures has facilitated research on various optical phenomena and applications by interfacing the near fields and free-propagating radiation. However, direct quantitative measurement of far-field scattering by a single nanostructure on the wavelength scale or less is highly challenging. Conventional back-focal-plane imaging covers only a limited solid angle determined by the numerical aperture of the objectives and suffers from optical aberration and distortion. Here, we present a quantitative measurement of the differential far-field scattering cross section of a single nanostructure over the full hemisphere. In goniometer-based far-field scanning with a high signal-to-noise ratio of approximately 27.4 dB, weak scattering signals are efficiently isolated and detected under total-internal-reflection illumination. Systematic measurements reveal that the total and differential scattering cross sections of a Au nanorod are determined by the plasmonic Fabry-Perot resonances and the phase-matching conditions to the free-propagating radiation, respectively. We believe that our angle-resolved far-field measurement scheme provides a way to investigate and evaluate the physical properties and performance of nano-optical materials and phenomena.