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Conical Frustum

\begin{figure}\BoxedEPSF{Frustum.epsf scaled 500}\end{figure}

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 top and bottom Radii. Then

\end{displaymath} (1)

The Surface Area, not including the top and bottom Circles, is
\end{displaymath} (2)

The Volume of the frustum is given by

V=\pi\int_0^h [r(z)]^2\,dz.
\end{displaymath} (3)

r(z)=R_1+(R_2-R_1){z\over h},
\end{displaymath} (4)

$\displaystyle V$ $\textstyle =$ $\displaystyle \pi \int_0^h \left[{R_1+(R_2-R_1){z\over h}}\right]^2\,dz$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 3}}\pi h({R_1}^2+R_1R_2+{R_2}^2).$ (5)

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={\textstyle{1\over 3}} h(A_1+A_2+\sqrt{A_1A_2}\,).
\end{displaymath} (6)

The weighted mean of $z$ over the frustum is
\left\langle{z}\right\rangle{}=\pi\int_0^h z[r(z)]^2\,dz={\textstyle{1\over 12}}h^2({R_1}^2+2R_1R_2+3{R_2}^2).
\end{displaymath} (7)

The Centroid is then located at a height
\bar z={\left\langle{z}\right\rangle{}\over V}={h({R_1}^2+2R_1R_2+3{R_2}^2)\over 4({R_1}^2+R_1R_2+{R_2}^2)}
\end{displaymath} (8)

(Beyer 1987, p. 133). The special case of the Cone is given by taking $R_2=0$, yielding $\bar z=h/4$.

See also Cone, Frustum, Pyramidal Frustum, Spherical Segment


Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 129-130 and 133, 1987.

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© 1996-9 Eric W. Weisstein