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
Volume element formula_25. In coordinates, where formula_27 is the determinant of the metric tensor "g" written in the coordinate system. A simple example of a volume element can be explored by considering a two-dimensional surface embedded in "n"-dimensional Euclidean space. Such a volume element is sometimes called an "area element". Consider a subset formula_28 and a mapping function thus defining a surface embedded in formula_30. In two dimensions, volume is just area, and a volume element gives a way to determine the area of parts of the surface. Thus a volume element is an expression of the form that allows one to