Forest compartments are usually delineated according to artificial or natural boundaries and usually include portions of different strata. While volume estimation of each stratum can be performed from field plots located within each stratum, volume estimation in portions of the stratum may be problematic owing to the small number (or even the absence) of plots falling in those portions. If upper canopy heights from airborne laser scanning are available at the pixel level for the whole survey area, these data are used as auxiliary information. A ratio model presuming a proportional relationship between transformed heights (e.g., power of heights) and volumes at the pixel level is adopted to guide estimation. From this model, the volume within any portion of the survey area is estimated as the proportionality factor estimate multiplied by the total of transformed heights in that portion. This estimator is considered from the model-based, design-based, and hybrid perspectives. Variances and their estimators are derived under the three approaches together with the corresponding confidence intervals. The volume estimator and the variance estimators are checked from the design-based point of view by a simulation study performed on a real forest in northwestern Italy. An application to a public forest estate in the same zone is performed.