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
Filling area conjecture onto , without increasing its volume. This implies that the replacement has not less volume than the original manifold . Filling area conjecture In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the surfaces that fill a closed curve of given length without introducing shortcuts between its points. Every smooth surface or curve in Euclidean space is a metric space, in which the (intrinsic) distance between two points of is defined as the infimum of the lengths of the curves that go from to "along" . For example, on a closed curve formula_1