# Cylinder

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A **cylinder** is one of the most curvilinear basic geometric shapes:It has two faces, zero vertices, and zero edges. 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

and its surface area is:

- the area of the top +
- the area of the bottom +
- the area of the side .

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

With the top and bottom, the surface area is

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

## Volume

Having a right circular cylinder with a height units and a base of radius units with the coordinate axes chosen so that the origin is at the center of one base and the height is measured along the positive x-axis. A plane section at a distance of units from the origin has an area of square units where

or

An element of volume, is a right cylinder of base area square units and a thickness of units. Thus if cubic units is the volume of the right circular cylinder, by Riemann sums,

## 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.^{[1]}

## Other types of cylinders

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

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.

An **oblique cylinder** has the top and bottom surfaces displaced from one another.

There are other more unusual types of cylinders. These are the *imaginary elliptic cylinders*:

the *hyperbolic cylinder*:

and the *parabolic cylinder*:

## Inscribed cylinder

The volume of a cylinder whose base is an inscribed circle is therefore:

The surface area of a cylinder whose base is an inscribed circle is therefore:

## Circumscribed cylinder

The cylinder with the base radius r and height 2r is known as a circumseribed cylinder whose volume is given as follows:

The surface area of a circumscribed cylinder is given as follows:

## References

## See also

- Steinmetz solid, the intersection of two or three perpendicular cylinders
- Prism (geometry)

## External links

Wikimedia Commons has media related to .Cylinder (geometry) |

- Surface area of a cylinder at MATHguide
- Volume of a cylinder at MATHguide
- Spinning Cylinder at Math Is Fun
- Volume of a cylinder Interactive animation at Math Open Reference

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