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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
be the slant height and
and
the base and top radii. Then



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(1)
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The surface area, not including the top and bottom circles, is
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(2)
|
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(3)
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The volume of the frustum is given by
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(4)
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But
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(5)
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so
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(6)
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(7)
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(8)
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This formula can be generalized to any pyramid by letting
be the base areas of the top and bottom of the frustum. Then the volume can be written as

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(9)
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The area-weighted integral of
over the frustum is

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(10)
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(11)
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so the geometric centroid is located along the z-axis at a height
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(12)
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(13)
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(Eshbach 1975, p. 453; Beyer 1987, p. 133; Harris and Stocker 1998, p. 105). The special case of the cone is given by taking
, yielding
.


Wolfram
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