Editing Cylinder (section)

==Properties==

===Cylindric sections===
[[Image:Cylindric section.svg|thumb|left|120px|Cylindric section]]

A cylindric section is the intersection of a cylinder's surface with a [[Plane (geometry)|plane]]. They are, in general, curves and are special types of [[cross section (geometry)|''plane sections'']]. The cylindric section by a plane that contains two elements of a cylinder is a [[parallelogram]].{{sfn|Wentworth|Smith|1913|p=354}} Such a cylindric section of a right cylinder is a [[rectangle]].{{sfn|Wentworth|Smith|1913|p=354}}

A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a ''{{dfn|right section}}''.{{sfn|Wentworth|Smith|1913|p=357}} If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a [[conic section]] (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively.
[[File:Blue cut-cylinder.gif|thumb|Cylindric sections of a right circular cylinder]]
For a right circular cylinder, there are several ways in which planes can meet a cylinder. First, planes that intersect a base in at most one point. A plane is tangent to the cylinder if it meets the cylinder in a single element.  The right sections are circles and all other planes intersect the cylindrical surface in an [[ellipse]].<ref>{{cite web |mode=cs2 |title=Cylindric section |website= [[MathWorld]] |url=http://mathworld.wolfram.com/CylindricSection.html }}</ref> If a plane intersects a base of the cylinder in exactly two points then the line segment joining these points is part of the cylindric section. If such a plane contains two elements, it has a rectangle as a cylindric section, otherwise the sides of the cylindric section are portions of an ellipse. Finally, if a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle.

In the case of a right circular cylinder with a cylindric section that is an ellipse, the [[Eccentricity (mathematics)|eccentricity]] {{math|''e''}} of the cylindric section and [[semi-major axis]] {{math|''a''}} of the cylindric section depend on the radius of the cylinder {{math|''r''}} and the angle {{math|''α''}} between the secant plane and cylinder axis, in the following way:
<math display="block">\begin{align}
e &= \cos\alpha, \\[1ex]
a &= \frac{r}{\sin\alpha}.
\end{align}</math>

===Volume===
If the base of a circular cylinder has a [[radius]]  {{math|''r''}} and the cylinder has height {{mvar|h}}, then its [[volume]] is given by
<math display=block>V = \pi r^2h</math>

This formula holds whether or not the cylinder is a right cylinder.{{sfn|Wentworth|Smith|1913|p=359}}

This formula may be established by using [[Cavalieri's principle]].
[[File:Elliptic cylinder abh.svg|thumb|A solid elliptic right cylinder with the semi-axes {{math|''a''}} and {{math|''b''}} for the base ellipse and height {{math|''h''}}]]
In more generality, by the same principle, the volume of any cylinder is the product of the area of a base and the height. For example, an elliptic cylinder with a base having [[Semi-major and semi-minor axes|semi-major axis]] {{mvar|a}}, semi-minor axis {{mvar|b}} and height {{mvar|h}} has a volume {{math|1=''V'' = ''Ah''}}, where {{mvar|A}} is the area of the base ellipse (= {{math|{{pi}}''ab''}}). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive {{mvar|x}}-axis and {{math|1=''A''(''x'') = ''A''}} the area of each elliptic cross-section, thus:
<math display=block>V = \int_0^h A(x) dx = \int_0^h \pi ab dx = \pi ab \int_0^h dx = \pi a b h.</math>

Using [[cylindrical coordinates]], the volume of a right circular cylinder can be calculated by integration
<math display=block>\begin{align}
V &= \int_0^h \int_0^{2\pi} \int_0^r s \,\, ds \, d\phi \, dz \\[5mu]
&= \pi\,r^2\,h.
\end{align}</math>

===Surface area===
Having radius {{math|''r''}} and altitude (height) {{mvar|h}}, the [[surface area]] of a right circular cylinder, oriented so that its axis is vertical, consists of three parts:

* the area of the top base: {{math|π''r''<sup>2</sup>}}
* the area of the bottom base: {{math|π''r''<sup>2</sup>}}
* the area of the side: {{math|2π''rh''}}

The area of the top and bottom bases is the same, and is called the ''base area'', {{math|''B''}}. The area of the side is known as the ''{{dfn|lateral area}}'', {{math|''L''}}.

An ''open cylinder'' does not include either top or bottom elements, and therefore has surface area (lateral area)
<math display=block>L = 2 \pi r h</math>

The surface area of the solid right circular cylinder is made up the sum of all three components: top, bottom and side. Its surface area is therefore
<math display=block>A = L + 2B = 2\pi rh + 2\pi r^2 = 2 \pi r (h + r) = \pi d (r + h)</math>
where {{math|1=''d'' = 2''r''}} is the [[diameter]] of the circular top or bottom.

For a given volume, the right circular cylinder with the smallest surface area has {{math|1=''h'' = 2''r''}}. Equivalently, for a given surface area, the right circular cylinder with the largest volume has {{math|1=''h'' = 2''r''}}, that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).<ref>{{citation |title=Calculus With Applications |first1=Peter D. |last1=Lax |author1-link=Peter Lax |first2=Maria Shea |last2=Terrell |publisher=Springer |year=2013 |isbn=9781461479468 |page=178 |url=https://books.google.com/books?id=dDq3BAAAQBAJ&pg=PA178 }}.</ref>

The lateral area, {{mvar|L}}, of a circular cylinder, which need not be a right cylinder, is more generally given by
<math display=block>L = e \times p,</math>
where {{mvar|e}} is the length of an element and {{mvar|p}} is the perimeter of a right section of the cylinder.{{sfn|Wentworth|Smith|1913|p=358}} This produces the previous formula for lateral area when the cylinder is a right circular cylinder.
[[File:Zylinder-rohr-s.svg|thumb|180px|Hollow cylinder]]

=== Right circular hollow cylinder (cylindrical shell)===

A ''right circular hollow cylinder'' (or ''{{dfn|cylindrical shell}}'') is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel [[Annulus (mathematics)|annular]] bases perpendicular to the cylinders' common axis, as in the diagram.

Let the height be {{math|''h''}}, internal radius {{math|''r''}}, and external radius {{math|''R''}}.  The volume is given by subtracting the volume of the inner imaginary cylinder (i.e. hollow space) from the volume of the outer cylinder:  
<math display=block>
V = \pi \left( R ^2 - r ^2 \right) h
= 2 \pi \left ( \frac{R + r}{2} \right) h (R - r).
</math>
Thus, the volume of a cylindrical shell equals {{nobr|2{{pi}}&thinsp;×}}&thinsp;{{nobr|average radius&thinsp;×}}&thinsp;{{nobr|height&thinsp;×}}&thinsp;thickness.{{sfn|Swokowski|1983|p=292}}

The surface area, including the top and bottom, is given by
<math display=block> A = 2 \pi \left( R + r \right) h + 2 \pi \left( R^2  - r^2 \right). </math>
Cylindrical shells are used in a common integration technique for finding volumes of solids of revolution.{{sfn|Swokowski|1983|p=291}}

===''On the Sphere and Cylinder''===

[[File:Esfera Arquímedes.svg|thumb|right|A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases]]
{{main|On the Sphere and Cylinder}}
In the treatise by this name, written {{Circa|225 BCE}}, [[Archimedes]] obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a [[sphere]] by exploiting the relationship between a sphere and its [[circumscribe]]d [[right circular cylinder]] of the same height and [[diameter]]. The sphere has a volume {{nowrap|two-thirds}} that of the circumscribed cylinder and a surface area  {{nowrap|two-thirds}} that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radius {{mvar|r}} is {{math|1={{sfrac|4|3}}{{pi}}''r''<sup>3</sup> = {{sfrac|2|3}} (2{{pi}}''r''<sup>3</sup>)}}. The surface area of this sphere is {{math|1=4{{pi}}''r''<sup>2</sup> = {{sfrac|2|3}} (6{{pi}}''r''<sup>2</sup>)}}. A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.