Surface area representing a cell as an idealized sphere of radius "r", the volume and surface area are, respectively, "V" = 4/3 π "r"; "SA" = 4 π "r". The resulting surface area to volume ratio is therefore 3/"r". Thus, if a cell has a radius of 1 μm, the SA:V ratio is 3; whereas if the radius of the cell is instead 10 μm, then the SA:V ratio becomes 0.3. With a cell radius of 100, SA:V ratio is 0.03. Thus, the surface area falls off steeply with increasing volume. Surface area The surface area of a solid object is a
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