# Cylinder (geometry)

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. The most common type of such generalized cylinders is given by certain quadric surfaces. A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an **elliptic cylinder**, **parabolic cylinder**, or **hyperbolic cylinder**.

## 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

- <math>V = \pi r^2 h \,</math>

and its surface area is:

- the area of the top <math>( \pi r^2 )\,</math> +
- the area of the bottom <math>( \pi r^2 )\,</math> +
- the area of the side <math>( 2 \pi r h )\,</math>.

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

- <math>A = 2 \pi r h.\,</math>

With the top and bottom, the surface area is

- <math>A = 2 \pi r^2 + 2 \pi r h = 2 \pi r ( r + h ).\,</math>

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.)

## Other types of cylinders

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

- <math>\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1.</math>

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*:

- <math>\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = -1</math>

the *hyperbolic cylinder*:

- <math>\left(\frac{x}{a}\right)^2 - \left(\frac{y}{b}\right)^2 = 1</math>

and the *parabolic cylinder*:

- <math>x^2 + 2ay = 0. \,</math>

## Trivia

- The volume of the cylinder is 3 times the volume of a cone with equal radius and equal height.

## See also

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

## External links

- 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 Referencear:أسطوانة (هندسة رياضية)

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