Consider a complex, highly convoluted three-dimensional object that has been digitized and is available as a set of voxels. How can one estimate the (original, continuous) area of a region of interest on the surface of the object? The problem appears in the analysis of segmented MRI brain data and in other three-dimensional imaging applications. Several di culties arise. First, due to the complexity of the surface and its foldings, the region of interest and its intended boundary can be concealed and are therefore di cult to delineate. Second, the correct surface topology on intricate voxel sets may not be obvious. Third, the surface area of a digital voxel world is generally very di erent than the area of the underlying continuous surface. These problems can be partly circumvented by transforming the voxel data to a polyhedral surface representation. Our challenge is to accomplish the task while maintaining the original voxel representation. Estimators for the continuous surface area of digital objects have been available for some time. However, the known methods are limited to fairly smooth and "well-behaved" surfaces. This research bridges the gap between the available surface area estimation theory, that applies to idealized settings, and the reality of MRI brain data. Surface connectivity ambiguities are alleviated by considering the object/background boundary voxel faces rather than the border voxels themselves. The region of interest on the surface is delimited by growing geodesic paths between user-provided anchor points. Surface estimation is extended to admit surfaces with higher curvature than previously considered. Performance evaluation results are provided, and operation on MRI brain data is demonstrated.