Volume as the disk radius, the volume of the sphere can be calculated as Now Combining yields formula_15 This formula can be derived more quickly using the formula for the sphere's surface area, which is formula_16. The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well. However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally thin circular disks of thickness
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