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14-September-2008 18:38:48 - Cylinder geometry A right circular cylinder A right circular cylinder 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. Contents 1 Common usage 2 Other types of cylinders 3 Trivia 4 See also 5 External links 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 = \pi r^2 h \, and its surface area is: the area of the top \pi r^2 \, + the area of the bottom \pi r^2 \, + the area of the side 2 \pi r h \, . Therefore without the top or bottom lateral area, the surface area is A = 2 \pi r h.\, With the top and bottom, the surface area is A = 2 \pi r^2 + 2 \pi r h = 2 \pi r r + h .\, 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. Other types of cylinders An elliptic cylinder An elliptic cylinder An elliptic cylinder is a quadric surface, with the following equation in Cartesian coordinates: \left\fracxa\right^2 + \left\fracyb\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. 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: \left\fracxa\right^2 + \left\fracyb\right^2 = -1 the hyperbolic cylinder: \left\fracxa\right^2 - \left\fracyb\right^2 = 1 and the parabolic cylinder: x^2 + 2ay = 0. \, 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 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 Retrieved from http://en..org/wiki/Cylinder_geometry Categories: Quadrics | Elementary geometry | Euclidean solid geometry Views Article Discussion this page History Personal tools Log in / create account Navigation Main page Contents Featured content Current events Random article Search Go Search Interaction Community portal Recent changes Contact Donate to Help Toolbox What links here Related changes Upload file Special pages Printable version Permanent link Cite this page Languages العربية Asturianu Aymar AzÉ™rbaycan Català Česky Dansk Deutsch Español Esperanto Euskara Français í•œêµì–´ Ido Bahasa Indonesia Ã?slenska Italiano עברית Lietuvių Magyar МакедонÑ?ки Nederlands 日本語 ‪Norsk bokmÃ¥l‬ Polski Português Română Runa Simi РуÑ?Ñ?кий Shqip Sicilianu Simple English SlovenÅ¡Ä?ina СрпÑ?ки / Srpski Basa Sunda Suomi Svenska ไทย Tiếng Việt Türkçe УкраїнÑ?ька ä¸æ–‡ This page was last modified on 6 September 2008, at 23:28
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