Solid angle surface area of a spherical cap is always equal to the area of a circle whose radius equals the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap. In the diagram this radius is given as: Hence for a unit sphere the solid angle of the spherical cap is given as: When = , the spherical cap becomes a hemisphere having a solid angle 2. The solid angle of the complement of the cone (picture a melon with the cone cut out) is clearly: This is also the solid
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