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Cylinder

From Simple English Wikipedia, the free encyclopedia
For other uses, seeCylinder (disambiguation).
A right circular cylinder

Acylinder is one of the most basic curved three dimensionalgeometric shapes, with thesurface formed by the points at a fixed distance from a givenline segment, known as theaxis of the cylinder. The shape can be thought of as acircularprism. Both the surface and the solid shape created inside can be called acylinder. Thesurface area and thevolume of a cylinder have been known since ancient times.

In differentialgeometry, a cylinder is defined more broadly as a ruled surface which is spanned by a one-parameter family ofparallel lines. A cylinder whosecross section is anellipse,parabola, orhyperbola is called anelliptic cylinder,parabolic cylinder, orhyperbolic cylinder respectively.

Common use

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In common use acylinder is taken to mean a finite section of aright circular cylinder, i.e., the cylinder with the generating lines perpendicular to the bases, with its ends closed to form two circular surfaces, as in the figure (right). If the cylinder has aradiusr{\displaystyle r} and length (height)h, then itsvolume is given by:

V=πr2h{\displaystyle V=\pi r^{2}h}

and itssurface area is:

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

A=2πrh{\displaystyle A=2\pi rh}.

With the top and bottom, the surface area is:

A=2πr2+2πrh=2πr(r+h){\displaystyle A=2\pi r^{2}+2\pi rh=2\pi r(r+h)}.

For a given volume, the cylinder with the smallest surface area hash=2r{\displaystyle h=2r}. For a given surface area, the cylinder with the largest volume hash=2r{\displaystyle h=2r}, i.e. the cylinder fits in a cube (height =diameter).

Volume

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Having a right circular cylinder with a heighth{\displaystyle h} units and a base of radiusr{\displaystyle r} 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 ofx{\displaystyle x} units from the origin has an area ofA(x){\displaystyle A(x)} square units where

A(x)=πr2{\displaystyle A(x)=\pi r^{2}}

or

A(y)=πr2{\displaystyle A(y)=\pi r^{2}}

An element of volume, is a right cylinder of base areaAwi{\displaystyle Aw^{i}} square units and a thickness ofΔix{\displaystyle \Delta _{i}x} units. Thus ifV cubic units is the volume of the right circular cylinder, byRiemann sums,

Volumeofcylinder=lim||Δ→0||∑i=1nA(wi)Δix{\displaystyle \mathrm {Volume\;of\;cylinder} =\lim _{||\Delta \to 0||}\sum _{i=1}^{n}A(w_{i})\Delta _{i}x}
=∫0hA(y)2dy{\displaystyle =\int _{0}^{h}A(y)^{2}\,dy}
=∫0hπr2dy{\displaystyle =\int _{0}^{h}\pi r^{2}\,dy}
=πr2h{\displaystyle =\pi \,r^{2}\,h\,}

Using cylindrical coordinates, the volume can be calculated by integration over

=∫0h∫02π∫0rsdsdϕdz{\displaystyle =\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{r}s\,\,ds\,d\phi \,dz}
=πr2h{\displaystyle =\pi \,r^{2}\,h\,}

Cylindric section

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Cylindric sections are the intersections of cylinders with planes. For a right circular cylinder, there are four possibilities. A plane tangent to the cylinder, meets the cylinder in a single straight line. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two parallel lines. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, in a circle.[1]

Other types of cylinders

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An elliptic cylinder

Anelliptic cylinder, or cylindroid, is a quadric surface, with the following equation inCartesian coordinates:

(xa)2+(yb)2=1.{\displaystyle ({\frac {x}{a}})^{2}+({\frac {y}{b}})^{2}=1.}

This equation is for anelliptic cylinder, a generalization of the ordinary,circular cylinder (a=b{\displaystyle a=b}). Even more general is thegeneralized cylinder: thecross-section can be any curve.

The cylinder is adegenerate quadric because at least one of the coordinates (in this casez{\displaystyle z}) does not appear in the equation.

Anoblique cylinder has the top and bottom surfaces displaced from one another.

There are other more unusual types of cylinders. These are theimaginary elliptic cylinders:

(xa)2+(yb)2=−1{\displaystyle ({\frac {x}{a}})^{2}+({\frac {y}{b}})^{2}=-1}

thehyperbolic cylinder:

(xa)2−(yb)2=1{\displaystyle ({\frac {x}{a}})^{2}-({\frac {y}{b}})^{2}=1}

and theparabolic cylinder:

x2+2ay=0.{\displaystyle x^{2}+2ay=0.\,}
Inprojective geometry, a cylinder is simply a cone whoseapex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.


References

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  1. ↑"MathWorld: Cylindric section".

Other websites

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Wikimedia Commons has media related toCylinder (geometry).
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