'roller', 'tumbler' has traditionally been a three-dimensional solid

On the Sphere and Cylinder

A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases

Main article: 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 two-thirds that of the circumscribed cylinder and a surface area two-thirds 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 r is 4/3πr³ = 2/3 (2πr³). The surface area of this sphere is 4πr² = 2/3 (6πr²). 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.^[12] Such cylinders have, at times, been referred to as generalized cylinders. Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder.^[13] Thus, this definition may be rephrased to say that a cylinder is any ruled surface spanned by a one-parameter family of parallel lines.
A cylinder having a right section that is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder and hyperbolic cylinder, respectively. These are degenerate quadric surfaces.^[14]

Parabolic cylinder
When the principal axes of a quadric are aligned with the reference frame (always possible for a quadric), a general equation of the quadric in three dimensions is given by with the coefficients being real numbers and not all of A, B and C being 0. If at least one variable does not appear in the equation, then the quadric is degenerate. If one variable is missing, we may assume by an appropriate rotation of axes that the variable z does not appear and the general equation of this type of degenerate quadric can be written as where