Consider a non-degenerate open convex cone C with vertex the origin in the n-dimensional Euclidean space E^n. We study volume properties of strictly convex hypersurfaces in the cone C. As a result, for example, if the volume of the region of an elliptic cone C cut off by the tangent hyperplane P of M at p is independent of the point p in M, then it is shown that the hypersurface M is part of an elliptic hyperboloid.