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A frustum of a decagonal pyramid
English Wikipedia has an article on:
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Borrowed from Latin frustum (morsel).


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frustum (plural frustums or frusta)

  1. A cone or pyramid whose tip has been truncated by a plane parallel to its base.
    • 1742, Colin MacLaurin, A Treatise of Fluxions, volume 1, page 25:
      In a parabolic conoid this difference vaniſhes, the fruſtum being always equal to a cylinder of the ſame height upon the ſection of the conoid that biſects the altitude of the fruſtum and is parallel to its baſes.
    • 1809, William Nicholson, “FRUSTUM”, in The British Encyclopedia, or Dictionary of Arts and Sciences; [], volume III (E … I), London: Printed by C[harles] Whittingham, []; for Longman, Hurst, Rees, and Orme, [], →OCLC:
      This theorem holds good for complete solids as well as frustums, whether right or oblique, and not only of the solids generated from the conic sections, but also of all pyramids, cones, and in short of any solid, whose parallel sections are similar figures.
    • 1974, Stanisław Lem, translated by Michael Kandel, The Cyberiad:
      Come, every frustum longs to be a cone,
      And every vector dreams of matrices.
    • 2006, Pawan Harish Nirnimesh, P. J. Narayanan, Culling an Object Hierarchy to a Frustum Hierarchy, Prem Kalra, Shmuel Peleg (editors), Computer Vision, Graphics and Image Processing: 5th Indian Conference, ICVGIP 2006, Springer, LNCS4338, page 252,
      However, when there are multiple view frustums (as in a tiled display wall), visibility culling time becomes substantial and cannot be hidden by pipelining it with other stages of rendering.
    • 2008, R. Benjamin Davis, Techniques to Assess Acoustic-structure Interaction in Liquid Rocket Engines, page 122:
      Here, the dynamics of the fluid-filled frusta of cones are considered (see Figure 5.5). The frusta are clamped at their roots and free at their ends.
  2. A portion of a sphere, or in general any solid, delimited by two parallel planes.
    • 1840, James Blundell, Observations on Some of the More Important Diseases of Women, page 131:
      In some women it[the os uteri] is flat, in many more tuberose, and forming, as it were, a frustum of a sphere; [] .
    • 2014, John Bird, Engineering Mathematics, page 183:
      Problem 22. Determine the volume of a frustum of a sphere of diameter 49.74 cm if the diameter[sic] of the ends of the frustum are 24.0 and 40.0 cm, and the height of the frustum is 7.00 cm.

Usage notes[edit]

The misspelling frustrum is by incorrect analogy with frustrate, also of Latin origin.[1]

  • (portion of a sphere): The portion of the surface of a sphere delimited by parallel planes (i.e., the curved surface of a frustum) may be called a zone; however, that term is also sometimes used as a synonym of frustum.

Derived terms[edit]

Related terms[edit]



  1. ^ Oxford English Dictionary, 1884–1928, and First Supplement, 1933.



From Proto-Italic *frustom, from Proto-Indo-European *bʰrus-tós, from *bʰrews- (to break up, cut).


The etymology is consistent with the /u/ in the first syllable being short, and the word is shown without a macron in De Vaan's dictionary.[1] Although Bennett 1907 says "ū acc. to the Romance",[2] there are related words in Romance that point to short u in the descendants of the derivative *frŭstiāre,[3] such as Old French froissier.


frū̆stum n (genitive frū̆stī); second declension

  1. a piece, bit; crumb, morsel, scrap of food


Second-declension noun (neuter).

Case Singular Plural
Nominative frū̆stum frū̆sta
Genitive frū̆stī frū̆stōrum
Dative frū̆stō frū̆stīs
Accusative frū̆stum frū̆sta
Ablative frū̆stō frū̆stīs
Vocative frū̆stum frū̆sta


Derived terms[edit]


  • English: frustum
  • Italian: frusta
  • Spanish: frusto, frusta
  • Vulgar Latin: *frustiāre (see there for further descendants)


  1. ^ De Vaan, Michiel (2008), “frustum”, in Etymological Dictionary of Latin and the other Italic Languages (Leiden Indo-European Etymological Dictionary Series; 7), Leiden, Boston: Brill, →ISBN, page 245
  2. ^ Charles E. Bennett (1907), “Hidden Quantity”, in The Latin Language – a historical outline of its sounds, inflections, and syntax, Boston: Allyn and Bacon, page 59
  3. ^ Walther von Wartburg (1928–2002), “*frustiare”, in Französisches Etymologisches Wörterbuch (in German), volume 3: D–F, page 831

Further reading[edit]

  • frustum”, in Charlton T. Lewis and Charles Short (1879) A Latin Dictionary, Oxford: Clarendon Press
  • frustum”, in Charlton T. Lewis (1891) An Elementary Latin Dictionary, New York: Harper & Brothers
  • frustum in Charles du Fresne du Cange’s Glossarium Mediæ et Infimæ Latinitatis (augmented edition with additions by D. P. Carpenterius, Adelungius and others, edited by Léopold Favre, 1883–1887)