Cylinder (geometry) - Citizendia

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. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. 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. Surface area is the measure of how much exposed Area an object has The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically

In differential geometry, a cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Geometry, a Surface S is ruled if through every point of S there is a straight line that lies on S The most common type of such generalized cylinders is given by certain quadric surfaces. In mathematics a quadric, or quadric surface, is any D -dimensional Hypersurface defined as the locus of zeros of a Quadratic A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions

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 $( 2 \pi r h )\,$ . Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically Surface area is the measure of how much exposed Area an object has

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

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

This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). In mathematics a quadric, or quadric surface, is any D -dimensional Hypersurface defined as the locus of zeros of a Quadratic In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane Even more general is the generalized cylinder: the cross-section can be any curve. In Geometry, a cross section is the intersection of a body in 2-dimensional space with a line or of a body in 3-dimensional space with a plane etc

The cylinder is a degenerate quadric because at least one of the coordinates (in this case z) does not appear in the equation. In mathematics a quadric, or quadric surface, is any D -dimensional Hypersurface defined as the locus of zeros of a Quadratic

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:

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

the hyperbolic cylinder:

$\left(\frac{x}{a}\right)^2 - \left(\frac{y}{b}\right)^2 = 1$

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.

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 Reference

In Geometry, the Steinmetz solid is the solid body generated by the intersection of two or three cylinders of equal radius at right angles General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces

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Contents

Common usage

Other types of cylinders

Trivia

See also

External links