Cylinder (geometry)

From Includipedia, the inclusionist encyclopedia

In mathematics, a 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. By some definitions the cylinder is not considered to be a quadric at all.

Contents 1 Common usage 2 Other types of cylinders 3 See also 4 External links

[edit] 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:

• Area of the top is <math>\pi r^2</math>
• Area of the bottom is <math>\pi r^2</math>
• Area of the side is <math>2 \pi r h</math>

Therefore without the top or bottom, 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 = 2r. For a given surface area, the cylinder with the largest volume has h = 2r, i.e. the cylinder fits in a sphere (height = diameter.)

[edit] Other types of cylinders

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>

[edit] See also

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

[edit] External links

• Surface Area MATHguide
• Volume MATHguide
• Spinning Cylinder Math Is Fun
• calculate surface area and volume with your own values
• Paper model cylinder
• Volume of a cylinder Interactive animationar:أسطوانة (هندسة رياضية)

az:Silindr ca:Cilindre cs:Válec da:Cylinder (geometri) de:Zylinder (Geometrie) es:Cilindro eo:Cilindro eu:Zilindro fr:Cylindre ko:원기둥 io:Cilindro id:Silinder is:Sívalningur it:Cilindro (geometria) he:גליל (גאומטריה) hu:Henger mk:Цилиндар (геометрија) nl:Cilinder ja:円柱 (数学) no:Sylinder pl:Walec (bryła) pt:Cilindro ro:Cilindru ru:Цилиндр sq:Cilindri simple:Cylinder sl:Valj sr:Ваљак (геометрија) fi:Lieriö sv:Cylinder th:ทรงกระบอก vi:Hình trụ tròn tr:Silindir zh:圆柱体