Mathwords logoMathwords

Cylindrical Segment — Definition, Formula & Examples

A cylindrical segment is the portion of a right circular cylinder cut off by a plane that is not parallel to the base, producing a solid with one circular face and one elliptical cross-section at the cut.

Given a right circular cylinder of radius rr and a cutting plane that intersects the cylinder such that the minimum height of the remaining solid is h1h_1 and the maximum height is h2h_2, the cylindrical segment is the enclosed solid bounded by the circular base, the lateral surface, and the oblique cutting plane.

Key Formula

V=12 πr2 (h1+h2)V = \frac{1}{2}\,\pi r^2\,(h_1 + h_2)
Where:
  • VV = Volume of the cylindrical segment
  • rr = Radius of the circular base of the cylinder
  • h1h_1 = Minimum height of the segment (shortest generator)
  • h2h_2 = Maximum height of the segment (longest generator)

How It Works

When a plane slices through a cylinder at an angle to the base, the resulting solid has a variable height that ranges from a minimum h1h_1 on one side to a maximum h2h_2 on the opposite side. By Cavalieri's principle, every horizontal cross-section of this segment is a full circle of radius rr, so the volume equals the base area times the average height. This makes the computation straightforward: you only need the radius and the two extreme heights.

Worked Example

Problem: A right circular cylinder has a base radius of 6 cm. A plane cuts through the cylinder so that the shortest side of the remaining segment is 4 cm and the tallest side is 10 cm. Find the volume of the cylindrical segment.
Identify values: The radius is 6 cm, the minimum height is 4 cm, and the maximum height is 10 cm.
r=6,h1=4,h2=10r = 6,\quad h_1 = 4,\quad h_2 = 10
Compute the average height: Average the two extreme heights.
hˉ=h1+h22=4+102=7 cm\bar{h} = \frac{h_1 + h_2}{2} = \frac{4 + 10}{2} = 7 \text{ cm}
Apply the volume formula: Multiply the base area by the average height.
V=πr2⋅hˉ=π(6)2(7)=252π≈791.7 cm3V = \pi r^2 \cdot \bar{h} = \pi(6)^2(7) = 252\pi \approx 791.7 \text{ cm}^3
Answer: The volume of the cylindrical segment is 252π≈791.7252\pi \approx 791.7 cm³.

Why It Matters

Cylindrical segments arise in engineering when pipes or tanks are cut at oblique angles for welding or fitting. The average-height formula provides a quick way to calculate displaced fluid volume in partially tilted cylindrical containers.

Common Mistakes

Mistake: Using only one height (either h1h_1 or h2h_2) instead of their average.
Correction: Because the cutting plane is oblique, the height varies linearly across the diameter. The correct volume uses the mean height 12(h1+h2)\frac{1}{2}(h_1 + h_2), which accounts for symmetric variation by Cavalieri's principle.