A cylindrical surface is 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. Any line in this family of parallel lines is called an element of the cylindrical surface. From a kinematics point of view, given a plane curve, called the directrix, a cylindrical surface is that surface traced out by a line, called the generatrix, not in the plane of the directrix, moving parallel to itself and always passing through the directrix. Any particular position of the generatrix is an element of the cylindrical surface.

A solid bounded by a cylindrical surface and two parallel planes is called a (solid) cylinder. The line segments determined by an element of the cylindrical surface between the two parallel planes is called an element of the cylinder. All the elements of a cylinder have equal lengths. The region bounded by the cylindrical surface in either of the parallel planes is called a base of the cylinder. The two bases of a cylinder are congruent figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a right cylinder, otherwise it is called an oblique cylinder. If the bases are disks (regions whose boundary is a circle) the cylinder is called a circular cylinder. In some elementary treatments, a cylinder always means a circular cylinder.

The height (or altitude) of a cylinder is the perpendicular distance between its bases.

The cylinder obtained by rotating a line segment about a fixed line that it is parallel to is a cylinder of revolution. A cylinder of revolution is a right circular cylinder. The height of a cylinder of revolution is the length of the generating line segment. The line that the segment is revolved about is called the axis of the cylinder and it passes through the centers of the two bases.

Surface area

Having radius r and altitude (height) h, the surface area of a right circular cylinder, oriented so that its axis is vertical, consists of three parts:

• the area of the top base: π r 2

• the area of the bottom base: π r 2

• the area of the side: 2 π r h