Searching a database of 3D-volume objects for objects which are similar to a given 3D search object is an important problem which arises in number of database applications — for example, in Medicine and CAD. In this paper, we present a new geometry-based solution to the problem of searching for similar 3D-volume objects. The problem is motivated from a real application in the medical domain where volume similarity is used as a basis for surgery decisions. Our solution for an efficient similarity search on large databases of 3D volume objects is based on a new geometric index structure. The basic idea of our new approach is to use the concept of hierarchical approximations of the 3D objects to speed up the search process. We formally show the correctness of our new approach and introduce two instantiations of our general idea, which are based on cuboid and octree approximations. We finally provide a performance evaluation of our new index structure revealing significant performance improvements over existing approaches.