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Frustum

From Wikipedia, the free encyclopedia
Portion of a solid that lies between two parallel planes cutting the solid
For other uses, seeFrustum (disambiguation).
Pentagonal frustum and square frustum

Ingeometry, afrustum (Latin for 'morsel';[a]pl. frusta orfrustums) is the portion of asolid (normally apyramid or acone) that lies between twoparallel planes cutting the solid. In the case of a pyramid, the base faces arepolygonal and the side faces aretrapezoidal. Aright frustum is aright pyramid or a right conetruncated perpendicularly to its axis;[3] otherwise, it is anoblique frustum.

In atruncated cone ortruncated pyramid, the truncation plane isnot necessarily parallel to the cone's base, as in a frustum.

If all its edges are the same length, then a frustum becomes aprism (possibly oblique or/and with irregular bases).

Elements, special cases, and related concepts

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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 theapex (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass ofprismatoids.

Two frusta with twocongruent bases joined at these congruent bases make abifrustum.

Formulas

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Volume

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Pyramidal frustum
Pyramidal frustum

The formula for the volume of a pyramidal square frustum was introduced by the ancientEgyptian mathematics in what is called theMoscow Mathematical Papyrus, written in the13th dynasty (c. 1850 BC):

V=h3(a2+ab+b2),{\displaystyle V={\frac {h}{3}}\left(a^{2}+ab+b^{2}\right),}

wherea andb are the base and top side lengths, andh 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.

Thevolume of a conical or pyramidal frustum is the volume of the solid before slicing its "apex" off, minus the volume of this "apex":

V=h1B1−h2B23,{\displaystyle V={\frac {h_{1}B_{1}-h_{2}B_{2}}{3}},}

whereB1 andB2 are the base and top areas, andh1 andh2 are the perpendicular heights from the apex to the base and top planes.

Considering that

B1h12=B2h22=B1B2h1h2=α,{\displaystyle {\frac {B_{1}}{h_{1}^{2}}}={\frac {B_{2}}{h_{2}^{2}}}={\frac {\sqrt {B_{1}B_{2}}}{h_{1}h_{2}}}=\alpha ,}

the formula for the volume can be expressed as the third of the product of this proportionality,α{\displaystyle \alpha }, and of thedifference of the cubes of the heightsh1 andh2 only:

V=h1αh12−h2αh223=αh13−h233.{\displaystyle V={\frac {h_{1}\alpha h_{1}^{2}-h_{2}\alpha h_{2}^{2}}{3}}=\alpha {\frac {h_{1}^{3}-h_{2}^{3}}{3}}.}

By using the identitya3 −b3 = (a −b)(a2 +ab +b2), one gets:

V=(h1−h2)αh12+h1h2+h223,{\displaystyle V=(h_{1}-h_{2})\alpha {\frac {h_{1}^{2}+h_{1}h_{2}+h_{2}^{2}}{3}},}

whereh1 −h2 =h is the height of the frustum.

Distributingα{\displaystyle \alpha } and substituting from its definition, theHeronian mean of areasB1 andB2 is obtained:

B1+B1B2+B23;{\displaystyle {\frac {B_{1}+{\sqrt {B_{1}B_{2}}}+B_{2}}{3}};}

the alternative formula is therefore:

V=h3(B1+B1B2+B2).{\displaystyle V={\frac {h}{3}}\left(B_{1}+{\sqrt {B_{1}B_{2}}}+B_{2}\right).}

Heron of Alexandria is noted for deriving this formula, and with it, encountering theimaginary unit: the square root of negative one.[4]

3D model of a conical frustum.

In particular:

  • The volume of a circular cone frustum is:
V=πh3(r12+r1r2+r22),{\displaystyle V={\frac {\pi h}{3}}\left(r_{1}^{2}+r_{1}r_{2}+r_{2}^{2}\right),}
wherer1 andr2 are the base and topradii.
  • The volume of a pyramidal frustum whose bases are regularn-gons is:
V=nh12(a12+a1a2+a22)cot⁡πn,{\displaystyle V={\frac {nh}{12}}\left(a_{1}^{2}+a_{1}a_{2}+a_{2}^{2}\right)\cot {\frac {\pi }{n}},}
wherea1 anda2 are the base and top side lengths.

Surface area

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Conical frustum

For a right circular conical frustum[5][6] theslant heights{\displaystyle s} is

s=(r1−r2)2+h2,{\displaystyle \displaystyle s={\sqrt {\left(r_{1}-r_{2}\right)^{2}+h^{2}}},}

the lateral surface area is

π(r1+r2)s,{\displaystyle \displaystyle \pi \left(r_{1}+r_{2}\right)s,}

and the total surface area is

π((r1+r2)s+r12+r22),{\displaystyle \displaystyle \pi \left(\left(r_{1}+r_{2}\right)s+r_{1}^{2}+r_{2}^{2}\right),}

wherer1 andr2 are the base and top radii respectively.

Examples

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See also

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Notes

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  1. ^The termfrustum comes from Latin frustum, meaning 'piece' or 'morsel'. The English word is often misspelled asfrustrum, a different Latin word cognate to the English word "frustrate".[1] The confusion between these two words is very old: a warning about them can be found in theAppendix Probi, and the works ofPlautus include a pun on them.[2]

References

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  1. ^Clark, John Spencer (1895).Teachers' Manual: Books I–VIII. For Prang's complete course in form-study and drawing, Books 7–8. Prang Educational Company. p. 49.
  2. ^Fontaine, Michael (2010).Funny Words in Plautine Comedy.Oxford University Press. pp. 117, 154.ISBN 9780195341447.
  3. ^Kern, William F.; Bland, James R. (1938).Solid Mensuration with Proofs. p. 67.
  4. ^Nahin, Paul.An Imaginary Tale: The story of√−1. Princeton University Press. 1998
  5. ^"Mathwords.com: Frustum". Retrieved17 July 2011.
  6. ^Al-Sammarraie, Ahmed T.; Vafai, Kambiz (2017). "Heat transfer augmentation through convergence angles in a pipe".Numerical Heat Transfer, Part A: Applications.72 (3): 197−214.Bibcode:2017NHTA...72..197A.doi:10.1080/10407782.2017.1372670.S2CID 125509773.

External links

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Look upfrustum in Wiktionary, the free dictionary.
Wikimedia Commons has media related toFrustums.
Platonic solids(regular)
Catalan solids
(duals of Archimedean)
Dihedral regular
Dihedral uniform
duals:
Dihedral others
Johnson solids
Pyramids,cupolae androtundae
Modifiedpyramids
Modifiedcupolae androtundae
Augmentedprisms
ModifiedPlatonic solids
ModifiedArchimedean solids
Otherelementary solids
(See alsoList of Johnson solids, a sortable table)
Degenerate polyhedra are initalics.
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