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
Surface area but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions. These properties uniquely characterize surface area for a wide class of geometric surfaces called "piecewise smooth". Such surfaces consist of finitely many pieces that can be represented in the parametric form with a continuously differentiable function formula_4 The area of an individual piece is defined by the formula Thus the area of "S" is obtained by integrating the length of the normal vector formula_6 to the surface over the appropriate region "D" in the parametric "uv" plane. The