Cylindrical Segment

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.

If the plane is titled 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. Then the height of the solid at distance is given by

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.,

(1)

(Harris and Stocker 1998, p. 103). The volume can be found directly through integration by noting that the height in polar and Cartesian coordinates is given by

			(2)
			(3)
			(4)

so

			(5)
			(6)
			(7)

The volume can also be computed by integrating over plane sections parallel to the -plane as

			(8)
			(9)
			(10)

Similarly, the volume-weighted coordinates are given by

			(11)
			(12)
			(13)

so the centroids are given by

			(14)
			(15)
			(16)

(cf. the strange parameterization used by Harris and Stocker 1998, p. 103).

Since the top cap is an ellipse with semimajor and semiminor axes

			(17)
			(18)
			(19)

its surface area

			(20)
			(21)
			(22)

(Harris and Stocker 1998, p. 103).

The lateral surface area is given by

			(23)
			(24)

(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.