frustum

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.

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}}):

:$V\; =\; \backslash frac\{h\}\{3\}\backslash left(a^2\; +\; ab\; +\; b^2\backslash right),$

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":

:$V\; =\; \backslash frac\{h\_1\; B\_1\; -\; h\_2\; B\_2\}\{3\},$

where {{math|B₁}} and {{math|B₂}} are the base and top areas, and {{math|h₁}} and {{math|h₂}} are the perpendicular heights from the apex to the base and top planes.

Considering that

:$\backslash frac\{B\_1\}\{h\_1^2\}\; =\; \backslash frac\{B\_2\}\{h\_2^2\}\; =\; \backslash frac\{\backslash sqrt\{B\_1B\_2\}\}\{h\_1h\_2\}\; =\; \backslash alpha,$

the formula for the volume can be expressed as the third of the product of this proportionality, $\backslash alpha$, and of the difference of the cubes of the heights {{math|h₁}} and {{math|h₂}} only:

:$V\; =\; \backslash frac\{h\_1\; \backslash alpha\; h\_1^2\; -\; h\_2\; \backslash alpha\; h\_2^2\}\{3\}\; =\; \backslash alpha\backslash frac\{h\_1^3\; -\; h\_2^3\}\{3\}.$

By using the identity {{math|1=a³ − b³ = (a − b)(a² + ab + b²)}}, one gets:

:$V\; =\; (h\_1\; -\; h\_2)\backslash alpha\backslash frac\{h\_1^2\; +\; h\_1h\_2\; +\; h\_2^2\}\{3\},$

where {{math|1=h₁ − h₂ = h}} is the height of the frustum.

Distributing $\backslash alpha$ and substituting from its definition, the Heronian mean of areas {{math|B₁}} and {{math|B₂}} is obtained:

:$\backslash frac\{B\_1\; +\; \backslash sqrt\{B\_1B\_2\}\; +\; B\_2\}\{3\};$

the alternative formula is therefore:

:$V\; =\; \backslash frac\{h\}\{3\}\backslash left(B\_1\; +\; \backslash sqrt\{B\_1B\_2\}\; +\; B\_2\backslash right).$

Heron of Alexandria is noted for deriving this formula, and with it, encountering the imaginary unit: the square root of negative one.Nahin, Paul. An Imaginary Tale: The story of {{sqrt|−1}}. Princeton University Press. 1998

In particular:

• The volume of a circular cone frustum is:

::$V\; =\; \backslash frac\{\backslash pi\; h\}\{3\}\backslash left(r\_1^2\; +\; r\_1r\_2\; +\; r\_2^2\backslash right),$

:where {{math|r₁}} and {{math|r₂}} are the base and top radii.

• The volume of a pyramidal frustum whose bases are regular {{mvar|n}}-gons is:

::$V\; =\; \backslash frac\{nh\}\{12\}\backslash left(a\_1^2\; +\; a\_1a\_2\; +\; a\_2^2\backslash right)\backslash cot\backslash frac\{\backslash pi\}\{n\},$

:where {{math|a₁}} and {{math|a₂}} are the base and top side lengths.

:Image:Frustum with symbols.svg

File:CroppedCone.svg

File:Tronco cono 3D.stl

For a right circular conical frustum{{cite web |url=http://www.mathwords.com/f/frustum.htm |title=Mathwords.com: Frustum |access-date=17 July 2011}}{{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 }} the slant height $s$ is

{{bi|left=1.6|$\backslash displaystyle\; s=\backslash sqrt\{\backslash left(r\_1-r\_2\backslash right)^2+h^2\},$}}

the lateral surface area is

{{bi|left=1.6|$\backslash displaystyle\; \backslash pi\backslash left(r\_1+r\_2\backslash right)s,$}}

and the total surface area is

{{bi|left=1.6|$\backslash displaystyle\; \backslash pi\backslash left(\backslash left(r\_1+r\_2\backslash right)s+r\_1^2+r\_2^2\backslash right),$}}

where r₁ and r₂ are the base and top radii respectively.

File:Rolo-Candies-US.jpg 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.
• 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.
• File:Garsų Gaudyklė, Gintaro ilanka, Neringa, Litva 01.jpg, Neringa, Lithuania]]Sound Catcher: a wooden structure in Lithuania.
• Valençay cheese
• Rolo candies

• Spherical frustum

{{Notelist}}

{{Reflist}}

{{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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