A cylinder (from Ancient Greek κύλινδρος (kúlindros) 'roller, tumbler')[1] has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.

A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology.

The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces.

The region bounded by the cylindrical surface in either of the parallel planes is called a base of the 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.

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.

The cylindric section by a plane that contains two elements of a cylinder is a parallelogram.

The right sections are circles and all other planes intersect the cylindrical surface in an ellipse.

In more generality, by the same principle, the volume of any cylinder is the product of the area of a base and the height.

Using cylindrical coordinates, the volume of a right circular cylinder can be calculated by integration

The surface area of the solid right circular cylinder is made up the sum of all three components: top, bottom and side.

Equivalently, for a given surface area, the right circular cylinder with the largest volume has h = 2r, that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).

Thus, the volume of a cylindrical shell equals 2π × average radius × altitude × thickness.

Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.

[11] 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.

In some areas of geometry and topology the term cylinder refers to what has been called a cylindrical surface.

[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.

has the same sign as the coefficients A and B, then the equation of an elliptic cylinder may be rewritten in Cartesian coordinates as:

Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid.

Finally, if AB = 0 assume, without loss of generality, that B = 0 and A = 1 to obtain the parabolic cylinders with equations that can be written as:[16]

In projective geometry, a cylinder is simply a cone whose apex (vertex) lies on the plane at infinity.

These cases give rise to the hyperbolic, parabolic or elliptic cylinders respectively.

A solid circular cylinder can be seen as the limiting case of a n-gonal prism where n approaches infinity.

The connection is very strong and many older texts treat prisms and cylinders simultaneously.

[18] One reason for the early emphasis (and sometimes exclusive treatment) on circular cylinders is that a circular base is the only type of geometric figure for which this technique works with the use of only elementary considerations (no appeal to calculus or more advanced mathematics).

Thus, for example, since a truncated prism is a prism whose bases do not lie in parallel planes, a solid cylinder whose bases do not lie in parallel planes would be called a truncated cylinder.

From a polyhedral viewpoint, a cylinder can also be seen as a dual of a bicone as an infinite-sided bipyramid.