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# Pyramidal Frustum

A pyramidal frustum is a frustum made by chopping the top off a pyramid. It is a special case of a prismatoid.

For a right pyramidal frustum, let be the slant height, the height, the bottom base perimeter, the top base perimeter, the bottom area, and the top area. Then the surface area (of the sides) and volume of a pyramidal frustum are given by

 (1) (2)

The geometric centroid of a right pyramidal frustum occurs at a height

 (3)

above the bottom base (Harris and Stocker 1998).

The bases of a right -gonal frustum are regular polygons of side lengths and with circumradii

 (4)

where is the side length, so the diagonal connecting corresponding vertices on top and bottom has length

 (5)

and the edge length is

 (6) (7)

The triangular () and square () right pyramidal frustums therefore have side surface areas

 (8) (9)

The area of a regular -gon is

 (10)

so the volumes of these frustums are

 (11) (12)

Conical Frustum, Frustum, Heronian Mean, Pyramid, Spherical Segment, Truncated Square Pyramid

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## References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 128, 1987.Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 3-4, 1990.Eves, H. A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, p. 7, 1965.Harris, J. W. and Stocker, H. "Frustum of a Pyramid." §4.3.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 99, 1998.Kern, W. F. and Bland, J. R. "Frustum of Regular Pyramid." §28 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 67-71, 1948.

## Referenced on Wolfram|Alpha

Pyramidal Frustum

## Cite this as:

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