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Prismatoid


Prismatoid

A prismatoid is a polyhedron having two polygons in parallel planes as bases and triangular or trapezoidal lateral faces with one side lying in one base and the opposite polyhedron vertex or side lying in the other base. Examples include the cube, cuboid, pyramidal frustum, prism, and pyramid.

Let A_1 be the area of the lower base, A_2 the area of the upper base, M the area of the midsection, and h the altitude. Then as first formulated by Ernst Ferdinand August,

 V=1/6h(A_1+4M+A_2)

(Kern and Bland 1948, pp. 76-77). This result is known as the prismatoid formula, or sometimes the prismatoid theorem. However, since the latter term is also used for a theorem about the decomposition of a prismatoid (Kern and Bland 1948, pp. 121-130), the term "theorem" is best not applied to the formula.


See also

Cuboid, General Prismatoid, Parallelepiped, Prismatoid Theorem, Prismoid, Pyramidal Frustum, Wedge

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References

Alsina, C. and Nelsen, R. B. A Mathematical Space Odyssey: Solid Geometry in the 21st Century. Providence, RI: Math. Assoc. Amer., p. 85, 2015.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 128 and 132, 1987.Halsted, G. B. Rational Geometry: A Textbook for the Science of Space. Based on Hilbert's Foundations, 2nd ed. New York: Wiley, 1907.Harris, J. W. and Stocker, H. "Prismoid, Prismatoid." §4.5.1 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 102, 1998.Kern, W. F. and Bland, J. R. "Prismatoid," "Prismatoid Theorem," "Proof of the Prismoidal Formula," and "Application of Prismatoid Theorem." §30 and 43-45 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 75-80 and 121-130, 1948.Meserve, B. E. and Pingry, R. E. "Some Notes on the Prismoidal Formula." Math. Teacher 45, 257-263, 1952.Welchons, A. M.; Krickenberger, W. R.; and Pearson, H. R. Solid Geometry. Boston: Ginn, pp. 274-275, 1959.

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Prismatoid

Cite this as:

Weisstein, Eric W. "Prismatoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Prismatoid.html

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