Basic geometrical characteristics of 3D objects may be estimated by counting or measuring intersections of the objects with randomized periodic patterns of manifolds. Grids of points can be used for volume measurement, grids of lines or curves can be used for volume and surface area measurements, grids of planes or surfaces can be used for all dimensional measurements. The measured object can be captured in 3D digital image, the randomized grid can be generated and the intersections can be automatically assessed and visualized by the computer together with the grid and the object so that the intersections can be interactively edited. The formulas for unbiased estimators using intersections with randomized spatial grids follow from general formulas of integral geometry. The variance of the estimators may depend on properties of both the grid and the measured objects such as extension (size, dimensional characteristics), distribution of orientations of tangent or normal spaces and on shape (curvatures). This makes design of efficient spatial grids a difficult task. Efficient grids usually fill the space uniformly and orientations of its tangents or normals to the grid fill the half sphere of orientations in a uniform manner. The formulas for variance employing the geometrical characteristics of the grids and objects will be presented. Spatial grids of points, lines, circles, cycloids, planes, cylinders and spheres will be demonstrated.