TOPICS
Search

Pyramidal Frustum


PyramidalFrustum1PyramidalFrustum2PyramidalFrustum3PyramidalFrustum4

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 s be the slant height, h the height, p_1 the bottom base perimeter, p_2 the top base perimeter, A_1 the bottom area, and A_2 the top area. Then the surface area (of the sides) and volume of a pyramidal frustum are given by

S=1/2(p_1+p_2)s
(1)
V=1/3h(A_1+A_2+sqrt(A_1A_2)).
(2)

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

 z^_=(h(A_1+2sqrt(A_1A_2)+3A_2))/(4(A_1+sqrt(A_1A_2)+A_2))
(3)

above the bottom base (Harris and Stocker 1998).

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

 R_n=1/2ccsc(pi/n),
(4)

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

 x_n=1/2(a-b)csc(pi/n),
(5)

and the edge length is

e_n=sqrt(d^2+h^2)
(6)
=sqrt(1/4csc^2(pi/n)(a-b)^2+h^2).
(7)

The triangular (n=3) and square (n=4) right pyramidal frustums therefore have side surface areas

S_3=3/2(a+b)sqrt(1/3(a-b)^2+h^2)
(8)
S_4=2(a+b)sqrt(1/2(a-b)^2+h^2).
(9)

The area of a regular n-gon is

 A_n=1/4nc^2cot(pi/n),
(10)

so the volumes of these frustums are

V_3=1/(12)sqrt(3)(a^2+ab+b^2)h
(11)
V_4=1/3(a^2+ab+b^2)h.
(12)

See also

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

Explore with Wolfram|Alpha

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

Subject classifications