the largest volume has , that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle). The lateral area, , of a circular cylinder, which need not be a right cylinder, is more generally given by: , where is the length of an element and is the perimeter of a right section of the cylinder. This produces the previous formula for lateral area when the cylinder is a right circular cylinder. Right circular hollow cylinder (cylindrical shell) A right circular hollow cylinder (or ) is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel annular bases perpendicular to the cylinders' common axis, as in the diagram. Let the height be , internal radius , and external radius . The volume is given by . Thus, the volume of a cylindrical shell equals 2(average radius)(altitude)(thickness). The surface area, including the top and bottom, is given by . Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution. On the Sphere and Cylinder In the treatise by this name, written c. 225 BCE, Archimedes obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a sphere by exploiting the relationship between a sphere and its circumscribed right circular cylinder of the same height and diameter. The sphere has a volume that of the circumscribed cylinder and a surface area that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radius is . The surface area of this sphere is . A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request. Cylindrical surfaces In some areas of geometry and topology the term cylinder refers to what has been called a cylindrical surface. A cylinder is defined as a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line. Such cylinders have, at times, been referred to as . Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder. Thus, this