Frustum: Difference between revisions

Latest revision as of 19:54, 4 November 2025

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In geometry, a Template:Langnf;Template:Efn (Template:Plural form: frusta or frustums) is the portion of a solid (normally a pyramid or a 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 truncated perpendicularly to its axis;^[1] otherwise, it is an oblique frustum.

In a truncated cone or truncated pyramid, the truncation plane is Template:Em necessarily parallel to the cone's base, as in a frustum.

If all its edges are the same length, then a frustum becomes a 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 (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of prismatoids.

Two frusta with two congruent bases joined at these congruent bases make a bifrustum.

Formulas

Volume

Pyramidal frustum

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 (Template:Circa):

V = \frac{h}{3} (a^{2} + a b + b^{2}),

where Template:Mvar and Template:Mvar are the base and top side lengths, and Template:Mvar 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":

V = \frac{h_{1} B_{1} - h_{2} B_{2}}{3},

where Template:Math and Template:Math are the base and top areas, and Template:Math and Template:Math are the perpendicular heights from the apex to the base and top planes.

Considering that

\frac{B_{1}}{h_{1}^{2}} = \frac{B_{2}}{h_{2}^{2}} = \frac{\sqrt{B_{1} B_{2}}}{h_{1} h_{2}} = α,

the formula for the volume can be expressed as the third of the product of this proportionality, $α$ , and of the difference of the cubes of the heights Template:Math and Template:Math only:

V = \frac{h_{1} α h_{1}^{2} - h_{2} α h_{2}^{2}}{3} = α \frac{h_{1}^{3} - h_{2}^{3}}{3} .

By using the identity Template:Math, one gets:

V = (h_{1} - h_{2}) α \frac{h_{1}^{2} + h_{1} h_{2} + h_{2}^{2}}{3},

where Template:Math is the height of the frustum.

Distributing $α$ and substituting from its definition, the Heronian mean of areas Template:Math and Template:Math is obtained:

\frac{B_{1} + \sqrt{B_{1} B_{2}} + B_{2}}{3};

the alternative formula is therefore:

V = \frac{h}{3} (B_{1} + \sqrt{B_{1} B_{2}} + B_{2}) .

Heron of Alexandria is noted for deriving this formula, and with it, encountering the imaginary unit: the square root of negative one.^[2]

File:Tronco cono 3D.stl

3D model of a conical frustum.

In particular:

The volume of a circular cone frustum is:

V = \frac{π h}{3} (r_{1}^{2} + r_{1} r_{2} + r_{2}^{2}),

where Template:Math and Template:Math are the base and top radii.

The volume of a pyramidal frustum whose bases are regular Template:Mvar-gons is:

V = \frac{n h}{12} (a_{1}^{2} + a_{1} a_{2} + a_{2}^{2}) \cot \frac{π}{n},

where Template:Math and Template:Math are the base and top side lengths.

Surface area

File:CroppedCone.svg

Conical frustum

For a right circular conical frustum^[3]^[4] the slant height $s$ is Template:Bi the lateral surface area is Template:Bi and the total surface area is Template:Bi where r₁ and r₂ are the base and top radii respectively.

Examples

File:Garsų Gaudyklė, Gintaro ilanka, Neringa, Litva 01.jpg
Sound Catcher, Neringa, Lithuania
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.
Ziggurats, step pyramids, and certain ancient 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 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".
Buckets and typical lampshades are everyday examples of conical frustums.
Drinking glasses and some space capsules are also some examples.
Sound Catcher: a wooden structure in Lithuania.
Valençay cheese
Rolo candies

Notes

Template:Notelist

References

Template:Reflist

External links

Template:Sister project Template:Sister project

Derivation of formula for the volume of frustums of pyramid and cone (Mathalino.com)
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Paper models of frustums (truncated pyramids)
Paper model of frustum (truncated cone)
Design paper models of conical frustum (truncated cones)

Template:Polyhedron navigator Template:Authority control

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Nahin, Paul. An Imaginary Tale: The story of
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Template:Rcat shell. Princeton University Press. 1998
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@@ Line 8: / Line 8: @@
 }}
-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 [[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).
+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 the same length, then a frustum becomes a [[Prism (geometry)|prism]] (possibly oblique or/and with irregular bases).
+{{Clear}}
 ==Elements, special cases, and related concepts==
@@ Line 20: / Line 24: @@
 Two frusta with two [[Congruence (geometry)|congruent]] bases joined at these congruent bases make a [[bifrustum]].
+{{Clear}}
 ==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}}):
+[[Image:Frustum with symbols.svg|Pyramidal frustum|alt=Pyramidal frustum|thumb]]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.
@@ Line 45: / Line 51: @@
 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>
+[[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>[[File:Tronco cono 3D.stl|thumb|3D model of a conical frustum.]]In particular:
-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>
@@ Line 55: / Line 59: @@
 ::<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>}}
@@ Line 67: / Line 70: @@
 {{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.
+{{Clear}}
 ==Examples==
-[[File:Rolo-Candies-US.jpg|thumb|[[Rolo]] brand chocolates approximate a right circular conic frustum, although not flat on top. ]]
+*[[File:Garsų Gaudyklė, Gintaro ilanka, Neringa, Litva 01.jpg|thumb|''[[Sound Catcher]]'', Neringa, Lithuania]]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]].
-*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]].
@@ Line 79: / Line 83: @@
 *[[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|''[[Sound Catcher]]'', Neringa, Lithuania]]''[[Sound Catcher]]'': a wooden structure in Lithuania.
+*''[[Sound Catcher]]'': a wooden structure in Lithuania.
 *[[Valençay cheese]]
 *[[Rolo]] candies

Frustum: Difference between revisions

Latest revision as of 19:54, 4 November 2025

Contents

Elements, special cases, and related concepts

Formulas

Volume

Surface area

Examples

See also

Notes

References

External links

Navigation menu

Frustum: Difference between revisions

Latest revision as of 19:54, 4 November 2025

Elements, special cases, and related concepts

Formulas

Volume

Surface area

Examples

See also

Notes

References

External links

Navigation menu

Search