A cylindrical segment, sometimes also called a truncated cylinder, is the solid cut from a circular cylinder by two (or more) planes.
If there are two cutting planes, one perpendicular to the axis of the cylinder and the other tilted with respect to it, the resulting solid is known as a cylindrical wedge.
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If the plane is tilted with respect to a circular cross section but does not cut the bottom base, the resulting cylindrical segment
has one circular cap and one elliptical cap (see above figure). Consider a cylinder
of radius 
and minimum and maximum heights 
and 
. Set up a coordinate system
with lower cap in the 
-plane, origin at the center
of the lower cap, and the 
-axis passing through the center
of the lower cap parallel to the projection
of the semimajor axis of the upper cap.
The volume of the cylindrical section can be obtained instantly by noting that two such sections can be fitted together to form a cylinder
of radius 
and height 
, so the volume of the original
wedge is half that of the cylinder of height 
, i.e.,
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(1)
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(Harris and Stocker 1998, p. 103). The volume can be found directly through integration by noting that the height in polar coordinates and Cartesian coordinates is given by
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(2)
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(3)
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(4)
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so
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(5)
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(6)
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(7)
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The volume can also be computed by integrating over plane sections parallel to the
-plane
as
 |  | ![2int_(-R)^Rsqrt(R^2-x^2)[x/(2R)(h_2-h_1)+1/2(h_1+h_2)]dx](/images/equations/CylindricalSegment/Inline30.svg) |
(8)
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 |  | ![int_(-R)^Rsqrt(R^2-x^2)[x/R(h_2-h_1)+(h_1+h_2)]dx](/images/equations/CylindricalSegment/Inline33.svg) |
(9)
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(10)
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Similarly, the volume-weighted coordinates are given by
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(11)
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(12)
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(13)
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so the geometric centroid has coordinates
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(14)
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(15)
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(16)
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(cf. the strange parameterization used by Harris and Stocker 1998, p. 103).
Let 
denote the angle between the upper cap and the lower cap.
Since the top cap is an ellipse with semimajor
axis and semiminor axis
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(17)
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(18)
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(19)
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its surface area
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(20)
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(21)
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(22)
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(Harris and Stocker 1998, p. 103).
The lateral surface area is given by
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(23)
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(24)
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(Harris and Stocker 1998, p. 103).
The solid cut from a horizontal cylinder of length 
and radius 
by a single plane oriented parallel
to the cylinder's axis
of symmetry (i.e., a portion of a horizontal cylindrical tank which is partially
filled with fluid) is called a horizontal
cylindrical segment. A common problem associated with this shape is the quarter-tank
problem, which is determination of the amount of gas needed to fill it one-quarter
full.
