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Conical Frustum

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ConicalFrustumGraphic
Frustum

A conical frustum is a frustum created by slicing the top off a cone (with the cut made parallel to the base). For a right circular cone, let s be the slant height and R_1 and R_2 the base and top radii. Then

s=sqrt((R_1-R_2)^2+h^2).
(1)

The surface area, not including the top and bottom circles, is

A=pi(R_1+R_2)s
(2)
=pi(R_1+R_2)sqrt((R_1-R_2)^2+h^2).
(3)

The volume of the frustum is given by

V=piint_0^h[r(z)]^2dz.
(4)

But

r(z)=R_1+(R_2-R_1)z/h,
(5)

so

V=piint_0^h[r(z)]^2dz
(6)
=piint_0^h[R_1+(R_2-R_1)z/h]^2dz
(7)
=1/3pih(R_1^2+R_1R_2+R_2^2).
(8)

This formula can be generalized to any pyramid by letting A_i be the base areas of the top and bottom of the frustum. Then the volume can be written as

V=1/3h(A_1+A_2+sqrt(A_1A_2)).
(9)

The area-weighted integral of z over the frustum is

<z>=piint_0^hz[r(z)]^2dz
(10)
=1/(12)pih^2(R_1^2+2R_1R_2+3R_2^2),
(11)

so the geometric centroid is located along the z-axis at a height

z^_=(<z>)/V
(12)
=(h(R_1^2+2R_1R_2+3R_2^2))/(4(R_1^2+R_1R_2+R_2^2))
(13)

(Eshbach 1975, p. 453; Beyer 1987, p. 133; Harris and Stocker 1998, p. 105). The special case of the cone is given by taking R_2=0, yielding z^_=h/4.

See also

Cone, Frustum, Pyramidal Frustum, Spherical Segment

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References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 129-130 and 133, 1987.Eshbach, O. W. Handbook of Engineering Fundamentals. New York: Wiley, 1975.Harris, J. W. and Stocker, H. "Frustum of a Right Circular Cone." §4.7.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 105, 1998.Kern, W. F. and Bland, J. R. "Frustum of Right Circular Cone." §29 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 71-75, 1948.

Cite this as:

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

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