Solid angle is much, much farther away. The solid angle subtended by the complete ()-dimensional spherical surface of the unit sphere in -dimensional Euclidean space can be defined in any number of dimensions . One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula where is the gamma function. When is an integer, the gamma function can be computed explicitly. It follows that This gives the expected results of 4 steradians for the 3D sphere bounded by a surface of area and 2 radians for the 2D circle bounded by a circumference of length
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