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| {{For|the summit in Antarctica|Mount Frustum}}
| | Electrical Engineer Arnulfo Pile from Rockglen, enjoys football, Silver Jewellery and maintain a journal. Loves to travel and was encouraged after making vacation to Henderson Island.<br><br>Here is my blog post; Hultquist Jewellery ([http://www.pinterest.com/lizzielanejewel/some-of-our-favourite-hultquist-necklaces-from-liz/ http://www.pinterest.com/lizzielanejewel/some-of-our-favourite-hultquist-necklaces-from-liz]) |
| {{infobox
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| |title = Set of pyramidal frustums
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| |image = [[File:Pentagonal frustum.svg|110px]][[Image:Usech kvadrat piramid.png|110px]]
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| |caption = Examples: Pentagonal and square frustum
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| |label1 = Faces
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| |data1 = ''n'' [[trapezoid]]s, 2 [[polygon|''n''-gons]]
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| |label2 = Edges
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| |data2 = 3''n''
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| |label3 = Vertices
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| |data3 = 2''n''
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| |label4 = [[List of spherical symmetry groups|Symmetry group]]
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| |data4 = [[Symmetry group#Three dimensions|C<sub>''n''v</sub>]], [1,''n''], (*''nn'')
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| |label5 = [[Dual polyhedron]]
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| |data6 =
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| |label6 = Properties
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| |data6 = convex}}
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| In [[geometry]], a '''frustum'''<ref>The term comes from [[Latin]] ''[[wikt:en:frustum#Latin|frustum]]'' meaning "piece" or "crumb". The English word is often misspelled as ''{{sic|hide=y|frus|trum}}'', probably because of a similarity with the common words "frustrate" and "[[frustration]]", also of Latin origin, or "[[wikt:fulcrum|fulcrum]]"</ref> (plural: ''frusta'' or ''frustums'') is the portion of a [[polyhedron|solid]] (normally a [[cone (geometry)|cone]] or [[pyramid (geometry)|pyramid]]) that lies between two [[parallel planes]] cutting it.
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| The term is commonly used in [[computer graphics]] to describe the three-dimensional region which is visible on the screen, the "[[viewing frustum]]", which is formed by a [[clipping (computer graphics)|clipped]] pyramid; in particular, ''[[frustum culling]]'' is a method of [[hidden surface determination]].
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| In the [[aerospace industry]], frustum is the common term for the [[payload fairing|fairing]] between two stages of a [[multistage rocket]] (such as the [[Saturn V]]), which is shaped like a [[truncation (geometry)|truncated]] cone.
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| ==Elements, special cases, and related concepts==
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| [[File:Square frustum.png|thumb|200px|Square frustum]]
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| [[File:Triangulated monorectified tetrahedron.png|thumb|A regular octahedron can be augmented on 3 faces to create a triangular frustum]]
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| Each plane section is a floor or base of the frustum. Its axis if any, is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise.
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| The height of a frustum is the perpendicular distance between the planes of the two bases.
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| Cones and pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the [[apex (geometry)|apex]] (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of the [[prismatoid]]s.
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| Two frusta joined at their bases make a [[bifrustum]].
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| ==Formulae==
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| ===Volume===
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| The volume formula of frustum of square pyramid was introduced by the ancient [[Egyptian mathematics]] in what is called the [[Moscow Mathematical Papyrus]], written ca. 1850 BC.:
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| :<math>V = \frac{1}{3} h(a^2 + a b +b^2).</math>
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| where ''a''<sub> </sub> and ''b''<sub> </sub> are the base and top side lengths of the truncated pyramid, and ''h''<sub> </sub> is the height.
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| The Egyptians knew the correct formula for obtaining the volume of a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus.
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| The [[volume]] of a conical or pyramidal frustum is the volume of the solid before slicing the apex off, minus the volume of the apex:
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| :<math>V = \frac{h_1 B_1 - h_2 B_2}{3}</math>
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| where ''B''<sub>1</sub> is the area of one base, ''B''<sub>2</sub> is the area of the other base, and ''h''<sub>1</sub>, ''h''<sub>2</sub> are the perpendicular heights from the apex to the planes of the two bases.
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| Considering that
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| :<math>\frac{B_1}{h_1^2}=\frac{B_2}{h_2^2}=\frac{\sqrt{B_1 B_2}}{h_1 h_2} =</math> <big>α</big>
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| the formula for the volume can be expressed as a product of this proportionality <big>α/3</big> and a [[Factorization#Sum.2Fdifference_of_two_cubes|difference of cubes]] of heights ''h''<sub>1</sub> and ''h''<sub>2</sub> only.
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| By factoring (''h''<sub>2</sub>−''h''<sub>1</sub>) = ''h'', the height of the frustum, and <big>α</big>(''h''<sub>1</sub><sup>2</sup> + ''h''<sub>1</sub>''h''<sub>2</sub> + ''h''<sub>2</sub><sup>2</sup>)/<big>3</big>, and distributing <big>α</big> and substituting from its definition, the [[Heronian mean]] of areas ''B''<sub>1</sub> and ''B''<sub>2</sub> is obtained. The alternative formula is therefore
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| :<math>V = \frac{h}{3}(B_1+\sqrt{B_1 B_2}+B_2)</math>
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| [[Heron of Alexandria]] is noted for deriving this formula and with it encountering the [[imaginary number]], the square root of negative one.<ref>Nahin, Paul. "An Imaginary Tale: The story of [the square root of minus one]." Princeton University Press. 1998</ref>
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| In particular, the volume of a circular cone frustum is
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| :<math>V = \frac{\pi h}{3}(R_1^2+R_1 R_2+R_2^2)</math>
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| where [[pi|''π'']] is 3.14159265..., and ''R''<sub>1</sub>, ''R''<sub>2</sub> are the [[Radius (geometry)|radii]] of the two bases.
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| [[Image:Frustum with symbols.svg|right|miniatur|x200px|Pyramidal frustum.]] | |
| The volume of a pyramidal frustum whose bases are ''n''-sided regular polygons is
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| :<math>V= \frac{n h}{12} (a_1^2+a_1a_2+a_2^2)\cot \frac{\pi}{n}</math>
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| where ''a''<sub>1</sub> and ''a''<sub>2</sub> are the sides of the two bases.
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| ===Surface area===
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| For a right circular conical frustum<ref>{{cite web |url=http://www.mathwords.com/f/frustum.htm |title=Mathwords.com: Frustum |accessdate=17 July 2011}}</ref>
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| :<math>\begin{align}\text{Lateral Surface Area}&=\pi(R_1+R_2)s\\
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| &=\pi(R_1+R_2)\sqrt{(R_1-R_2)^2+h^2}\end{align}</math>
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| and
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| :<math>\begin{align}\text{Total Surface Area}&=\pi((R_1+R_2)s+R_1^2+R_2^2)\\
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| &=\pi((R_1+R_2)\sqrt{(R_1-R_2)^2+h^2}+R_1^2+R_2^2)\end{align}</math>
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| where ''R''<sub>1</sub> and ''R''<sub>2</sub> are the base and top radii respectively, and ''s'' is the slant height of the frustum.
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| The surface area of a right frustum whose bases are similar regular ''n''-sided [[polygon]]s is
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| :<math>A= \frac{n}{4}\left[(a_1^2+a_2^2)\cot \frac{\pi}{n} + \sqrt{(a_1^2-a_2^2)^2\sec^2 \frac{\pi}{n}+4 h^2(a_1+a_2)^2} \right]</math>
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| where ''a''<sub>1</sub> and ''a''<sub>2</sub> are the sides of the two bases.
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| == Examples ==
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| *On the back (the reverse) of a [[United States one-dollar bill]], a pyramidal frustum appears on the reverse of the [[Great Seal of the United States]], surmounted by the [[Eye of Providence]].
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| *Certain ancient [[Indigenous peoples of the Americas|Native American]] mounds also form the frustum of a pyramid.
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| *[[Chinese pyramids]].
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| *The [[John Hancock Center]] in [[Chicago]], [[Illinois]] is a frustum whose bases are rectangles.
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| *The [[Washington Monument]] is a narrow square-based pyramidal frustum topped by a small pyramid.
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| *The [[viewing frustum]] in [[3D computer graphics]] is a virtual photographic or video camera's usable [[field of view]] modeled as a pyramidal frustum.
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| *In the [[English language|English]] translation of [[Stanislaw Lem]]'s short-story collection ''[[The Cyberiad]]'', the poem ''Love and [[tensor algebra]]'' claims that "every frustum longs to be a cone".
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| *A [[bucket]] is an everyday example of a conical frustum. The internal diameter of its bottom is usually smaller than of its top.
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| *A typical [[lampshade]] is a frustum, as is the shape of a [[Fez (clothing)|fez]].
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| <!--Not so much an example as part of the frustrum's history?:
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| *The ancient Egyptians determined the equation for the volume of a frustum, as seen in the [[Moscow Mathematical Papyrus#Problem 14: Volume of frustum of square pyramid|Moscow Mathematical Papyrus]]
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| --> | |
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| ==Notes==
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| <references/>
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| ==External links==
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| {{Wiktionary|frustum}}
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| {{Commons category|Frustums}}
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| *[http://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-volume-of-a-frustum Derivation of formula for the volume of frustums of pyramid and cone] (Mathalino.com)
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| *{{MathWorld |urlname=PyramidalFrustum |title=Pyramidal frustum}}
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| *{{MathWorld |urlname=ConicalFrustum |title=Conical frustum}}
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| *[http://www.korthalsaltes.com/model.php?name_en=truncated%20pyramids%20of%20the%20same%20height Paper models of frustums (truncated pyramids)]
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| *[http://www.korthalsaltes.com/model.php?name_en=tapared%20cylinder Paper model of frustum (truncated cone)]
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| *[http://www.verbacom.com/cone/cone.php Design paper models of conical frustum (truncated cones)]
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| {{Polyhedron navigator}}
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| [[Category:Polyhedra]]
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| [[Category:Prismatoid polyhedra]]
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