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

Right circular cylinders
The bare term cylinder often refers to a solid cylinder with circular ends perpendicular to the axis, that is, a right circular cylinder, as shown in the figure. The cylindrical surface without the ends is called an open cylinder. The formulae for the surface area and the volume of a right circular cylinder have been known from early antiquity.
A right circular cylinder can also be thought of as the solid of revolution generated by rotating a rectangle about one of its sides. These cylinders are used in an integration technique (the "disk method") for obtaining volumes of solids of revolution.^[3]
A tall and thin needle cylinder has a height much greater than its diameter, whereas a short and wide disk cylinder has a diameter much greater than its height.
Properties
Cylindric sections

Cylindric section
A cylindric section is the intersection of a cylinder's surface with a plane. They are, in general, curves and are special types of plane sections. The cylindric section by a plane that contains two elements of a cylinder is a parallelogram.^[4] Such a cylindric section of a right cylinder is a rectangle.^[4]
A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a right section.^[5] If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a conic section (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively.

Cylindric sections of a right circular cylinder
For a right circular cylinder, there are several ways in which planes can meet a cylinder. First, planes that intersect a base in at most one point. A plane is tangent to the cylinder if it meets the cylinder in a single element. The right sections are circles and all other planes intersect the cylindrical surface in an ellipse.^[6] If a plane intersects a base of the cylinder in exactly two points then the line segment joining these points is part of the cylindric section. If such a plane contains two elements, it has a rectangle as a cylindric section, otherwise the sides of the cylindric section are portions of an ellipse. Finally, if a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle.
In the case of a right circular cylinder with a cylindric section that is an ellipse, the eccentricity e of the cylindric section and semi-major axis a of the cylindric section depend on the radius of the cylinder r and the angle α between the secant plane and cylinder axis, in the following way:
Volume
If the base of a circular cylinder has a radius r and the cylinder has height h, then its volume is given by
V = πr²h.
This formula holds whether or not the cylinder is a right cylinder.^[7]
This formula may be established by using Cavalieri's principle.

A solid elliptic cylinder with the semi-axes a and b for the base ellipse and height h
In more generality, by the same principle, the volume of any cylinder is the product of the area of a base and the height. For example, an elliptic cylinder with a base having semi-major axis a, semi-minor axis b and height h has a volume V = Ah, where A is the area of the base ellipse (= πab). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive x-axis and A(x) = A the area of each elliptic cross-section, thus:
Using cylindrical coordinates, the volume of a right circular cylinder can be calculated by integration over