Let be the area
of the lower base,
the area of the upper base, the area of the midsection, and the altitude.
Then as first formulated by Ernst Ferdinand August,
(Kern and Bland 1948, pp. 76-77). This is the prismatoid formula, also called the prismatoid theorem in some sources. A different prismatoid
theorem decomposes a prismatoid into a pyramid, a
wedge, and a parallelepiped
(Kern and Bland 1948, pp. 121-130).
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. Teacher45, 257-263, 1952.Welchons,
A. M.; Krickenberger, W. R.; and Pearson, H. R. Solid Geometry.
Boston, MA: Ginn, pp. 274-275, 1959.
be the area
of the lower base, 
the area of the upper base, 
the area of the midsection, and 
the altitude.
Then as first formulated by Ernst Ferdinand August,
