Solid angle to the unit sphere with center P, which can be calculated as the surface integral: where formula_5 is the unit vector corresponding to formula_6, the position vector of an infinitesimal area of surface with respect to point P, and where formula_7 represents the unit normal vector to . Even if the projection on the unit sphere to the surface S is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product formula_8. Thus one can approximate the solid angle subtended by a small facet having flat surface area ,
Gabriel's Horn diameter of the throat of the horn. The converse phenomenon of Gabriel's horn – a surface of revolution that has a "finite" surface area but an "infinite" volume – cannot occur: Let be a continuously differentiable function. Write for the solid of revolution of the graph about the -axis. "If the surface area of is finite, then so is the volume." Since the lateral surface area is finite, the limit superior: Therefore, there exists a such that the supremum } is finite. Hence, Finally, the volume: Therefore: "if the area is finite, then the volume must also be finite." Gabriel's