Nanoscale optical materials are of great interest for building future optoelectronic devices for information processing and sensing applications. Although heat transfer ultimately limits the maximum power at which nanoscale devices may operate, gaining a quantitative experimental measurement of photothermal heating within single nanostructures remains a challenge. Here, we measure the nonlinear optical absorption coefficient of optically trapped cadmium-sulfide nanoribbons at the level of single nanostructures through observations of their Brownian dynamics during single-beam laser trapping experiments. A general solution to the heat transfer partial differential equation is derived for nanostructures having rectilinear morphology including nanocubes and nanoribbons. Numerical electromagnetic calculations using the discrete-dipole approximation enable the simulation of the photothermal heating source function and the extraction of nonlinear optical absorption coefficients from experimental observations of single nanoribbon dynamics.