View source for Frustum

{{Short description|Portion of a solid that lies between two parallel planes cutting the solid}}
{{other uses}}
{{multiple image
 | image1 = Pentagonal frustum.svg
 | image2 = Usech kvadrat piramid.png
 | total_width = 450
 | footer = Pentagonal frustum and square frustum
}}

In [[geometry]], a {{langnf|la|'''frustum'''|italic=no|morsel}};{{efn|The term ''frustum'' comes {{etymology|la|{{wikt-lang|la|frustum}}|}}, meaning 'piece' or 'morsel". The English word is often misspelled as ''{{sic|hide=y|frus|trum}}'', a different Latin word cognate to the English word "frustrate".<ref>{{cite book |title=Teachers' Manual: Books I–VIII. For Prang's complete course in form-study and drawing, Books 7–8 |first=John Spencer|last=Clark |publisher=Prang Educational Company |year=1895 |page=49 |url=https://books.google.com/books?id=83EBAAAAYAAJ&pg=PA49}}</ref> The confusion between these two words is very old: a warning about them can be found in the ''[[Appendix Probi]]'', and the works of [[Plautus]] include a pun on them.<ref>{{cite book |title=Funny Words in Plautine Comedy |first=Michael|last=Fontaine |publisher=[[Oxford University Press]] |year=2010 |isbn=9780195341447 |url=https://books.google.com/books?id=SFPUvjlSUIsC&pg=PA117 |pages=117, 154}}</ref>}} ({{plural form}}: '''frusta''' or '''frustums''') is the portion of a [[polyhedron|solid]] (normally a [[pyramid (geometry)|pyramid]] or a [[cone (geometry)|cone]]) that lies between two [[parallel planes]] cutting the solid. In the case of a pyramid, the base faces are [[polygonal]] and the side faces are [[trapezoidal]]. A '''''right frustum''''' is a [[right pyramid]] or a right cone [[truncation (geometry)|truncated]] perpendicularly to its axis;<ref>{{cite book |first1=William F.|last1=Kern |first2=James R.|last2=Bland |title=Solid Mensuration with Proofs |year=1938 |page=67}}</ref> otherwise, it is an '''''oblique frustum'''''. 
In a ''[[truncated cone]]'' or ''[[truncated pyramid]]'', the truncation plane is {{em|not}} necessarily parallel to the cone's base, as in a frustum.
If all its edges are forced to become of the same length, then a frustum becomes a ''[[Prism (geometry)|prism]]'' (possibly oblique or/and with irregular bases).

==Elements, special cases, and related concepts==
A frustum's axis 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.

The height of a frustum is the perpendicular distance between the planes of the two bases.

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 [[prismatoid]]s.

Two frusta with two [[Congruence (geometry)|congruent]] bases joined at these congruent bases make a [[bifrustum]].

==Formulas==
===Volume===
The formula for the volume of a pyramidal square frustum was introduced by the ancient [[Egyptian mathematics]] in what is called the [[Moscow Mathematical Papyrus]], written in the [[13th dynasty]] ({{circa|1850 BC}}):
:<math>V = \frac{h}{3}\left(a^2 + ab + b^2\right),</math>
where {{mvar|a}} and {{mvar|b}} are the base and top side lengths, and {{mvar|h}} is the height.

The Egyptians knew the correct formula for the volume of such a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus.

The [[volume]] of a conical or pyramidal frustum is the volume of the solid before slicing its "apex" off, minus the volume of this "apex":
:<math>V = \frac{h_1 B_1 - h_2 B_2}{3},</math>
where {{math|''B''<sub>1</sub>}} and {{math|''B''<sub>2</sub>}} are the base and top areas, and {{math|''h''<sub>1</sub>}} and {{math|''h''<sub>2</sub>}} are the perpendicular heights from the apex to the base and top planes.

Considering that
:<math>\frac{B_1}{h_1^2} = \frac{B_2}{h_2^2} = \frac{\sqrt{B_1B_2}}{h_1h_2} = \alpha,</math>
the formula for the volume can be expressed as the third of the product of this proportionality, <math>\alpha</math>, and of the [[Factorization#Sum/difference of two cubes|difference of the cubes]] of the heights {{math|''h''<sub>1</sub>}} and {{math|''h''<sub>2</sub>}} only:
:<math>V = \frac{h_1 \alpha h_1^2 - h_2 \alpha h_2^2}{3} = \alpha\frac{h_1^3 - h_2^3}{3}.</math>
By using the identity {{math|1=''a''<sup>3</sup> − ''b''<sup>3</sup> = (''a'' − ''b'')(''a''<sup>2</sup> + ''ab'' + ''b''<sup>2</sup>)}}, one gets:
:<math>V = (h_1 - h_2)\alpha\frac{h_1^2 + h_1h_2 + h_2^2}{3},</math>
where {{math|1=''h''<sub>1</sub> − ''h''<sub>2</sub> = ''h''}} is the height of the frustum.

Distributing <math>\alpha</math> and substituting from its definition, the [[Heronian mean]] of areas {{math|''B''<sub>1</sub>}} and {{math|''B''<sub>2</sub>}} is obtained:
:<math>\frac{B_1 + \sqrt{B_1B_2} + B_2}{3};</math>
the alternative formula is therefore:
:<math>V = \frac{h}{3}\left(B_1 + \sqrt{B_1B_2} + B_2\right).</math>
[[Heron of Alexandria]] is noted for deriving this formula, and with it, encountering the [[imaginary unit]]: the square root of negative one.<ref>Nahin, Paul. ''An Imaginary Tale: The story of {{sqrt|−1}}.'' Princeton University Press. 1998</ref>

In particular:
*The volume of a circular cone frustum is:
::<math>V = \frac{\pi h}{3}\left(r_1^2 + r_1r_2 + r_2^2\right),</math>
:where {{math|''r''<sub>1</sub>}} and {{math|''r''<sub>2</sub>}} are the base and top [[Radius (geometry)|radii]].

*The volume of a pyramidal frustum whose bases are regular {{mvar|n}}-gons is:
::<math>V = \frac{nh}{12}\left(a_1^2 + a_1a_2 + a_2^2\right)\cot\frac{\pi}{n},</math>
:where {{math|''a''<sub>1</sub>}} and {{math|''a''<sub>2</sub>}} are the base and top side lengths.
:[[Image:Frustum with symbols.svg|right|Pyramidal frustum|frameless]]

===Surface area===
[[File:CroppedCone.svg|thumb|Conical frustum]]
[[File:Tronco cono 3D.stl|thumb|3D model of a conical frustum.]]
For a right circular conical frustum<ref>{{cite web |url=http://www.mathwords.com/f/frustum.htm |title=Mathwords.com: Frustum |access-date=17 July 2011}}</ref><ref>{{cite journal|doi=10.1080/10407782.2017.1372670 |first1=Ahmed T. |last1=Al-Sammarraie |first2=Kambiz |last2=Vafai |date=2017 |title=Heat transfer augmentation through convergence angles in a pipe |journal=Numerical Heat Transfer, Part A: Applications |volume=72 |issue=3 |page=197−214|bibcode=2017NHTA...72..197A |s2cid=125509773 }}</ref> the [[slant height]] <math>s</math> is
{{bi|left=1.6|<math>\displaystyle s=\sqrt{\left(r_1-r_2\right)^2+h^2},</math>}}
the lateral surface area is
{{bi|left=1.6|<math>\displaystyle \pi\left(r_1+r_2\right)s,</math>}}
and the total surface area is
{{bi|left=1.6|<math>\displaystyle \pi\left(\left(r_1+r_2\right)s+r_1^2+r_2^2\right),</math>}}
where ''r''<sub>1</sub> and ''r''<sub>2</sub> are the base and top radii respectively.

==Examples==
[[File:Rolo-Candies-US.jpg|thumb|[[Rolo]] brand chocolates approximate a right circular conic frustum, although not flat on top. ]]
*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]].
*[[Ziggurat]]s, [[step pyramid]]s, and certain ancient [[Indigenous peoples of the Americas|Native American]] mounds also form the frustum of one or more pyramids, with additional features such as stairs added.
*[[Chinese pyramids]].
*The [[John Hancock Center]] in [[Chicago]], [[Illinois]] is a frustum whose bases are rectangles.
*The [[Washington Monument]] is a narrow square-based pyramidal frustum topped by a small pyramid.
*The [[viewing frustum]] in [[3D computer graphics]] is a virtual photographic or video camera's usable [[field of view]] modeled as a pyramidal frustum.
*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".
*[[Bucket]]s and typical [[lampshade]]s are everyday examples of conical frustums.
*Drinking glasses and some [[space capsule]]s are also some examples.
*[[File:Garsų Gaudyklė, Gintaro ilanka, Neringa, Litva 01.jpg|thumb|{{ill|Garsų Gaudyklė|cs|Garsų gaudyklė}} , Neringa, Lithuania]][[Garsų Gaudyklė]]: a wooden structure in Lithuania.
*[[Valençay cheese]]
*[[Rolo]] candies

==See also==
*[[Spherical frustum]]

==Notes==
{{Notelist}}

==References==
{{Reflist}}

==External links==
{{Wiktionary|frustum}}
{{Commons category|Frustums}}
*[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)
*{{MathWorld |urlname=PyramidalFrustum |title=Pyramidal frustum}}
*{{MathWorld |urlname=ConicalFrustum |title=Conical frustum}}
*[http://www.korthalsaltes.com/model.php?name_en=truncated%20pyramids%20of%20the%20same%20height Paper models of frustums (truncated pyramids)]
*[http://www.korthalsaltes.com/model.php?name_en=tapared%20cylinder Paper model of frustum (truncated cone)]
*[http://www.verbacom.com/cone/cone.php Design paper models of conical frustum (truncated cones)]

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[[Category:Polyhedra]]
[[Category:Prismatoid polyhedra]]