A cylinder is one of the most basic curvilinear geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity.

In differential geometry, a cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder. A prism is a cylinder whose cross-section is a polygon.

Common usage

In common usage, a cylinder is taken to mean a finite section of a right circular cylinder with its ends closed to form two circular surfaces, as in the figure (right). If the cylinder has a radius r and length (height) h, then its volume is given by

$V\; =\; \backslash pi\; r^2\; h\; \backslash ,$

and its surface area is:

• the area of the top $(\; \backslash pi\; r^2\; )\backslash ,$ +
• the area of the bottom $(\; \backslash pi\; r^2\; )\backslash ,$ +
• the area of the side $(\; 2\; \backslash pi\; r\; h\; )\backslash ,$.

Therefore without the top or bottom (lateral area), the surface area is

$A\; =\; 2\; \backslash pi\; r\; h.\backslash ,$

With the top and bottom, the surface area is

$A\; =\; 2\; \backslash pi\; r^2\; +\; 2\; \backslash pi\; r\; h\; =\; 2\; \backslash pi\; r\; (\; r\; +\; h\; ).\backslash ,$

For a given volume, the cylinder with the smallest surface area has h = 2r. For a given surface area, the cylinder with the largest volume has h = 2r, i.e. the cylinder fits in a cube (height = diameter.)

Cylindric section

Cylindric sections are the intersections of cylinders with planes. Although these mostly yield ellipses (or circles), a degenerate case of two parallel lines, known as a ribbon, can also be produced, and it is also possible for there to be no intersection at all.

Other types of cylinders

An elliptic cylinder is a quadric surface, with the following equation in Cartesian coordinates:

$\backslash left(\backslash frac\{x\}\{a\}\backslash right)^2\; +\; \backslash left(\backslash frac\{y\}\{b\}\backslash right)^2\; =\; 1.$

This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). Even more general is the generalized cylinder: the cross-section can be any curve.

The cylinder is a degenerate quadric because at least one of the coordinates (in this case z) does not appear in the equation.