Truncated Cylinder or Prism

Truncated Cylinder or Prism

A cylinder or prism which has one base cut off by an intersecting plane. The other base is unaffected by the truncation. The truncating plane may be either oblique or parallel to the bases.

3D truncated prism with rectangular base, vertical edges, and a slanted top face cutting diagonally across the upper portion.

See also

Truncated cone or pyramid

Key Formula

V = A_{\text{base}} \times h_{\text{avg}}

Where:

$V$ = Volume of the truncated cylinder or prism
$A_{\text{base}}$ = Area of the intact (uncut) base
$h_{\text{avg}}$ = Average of the heights measured from the base to the truncating plane, taken at corresponding points (or, for a cylinder, the average of the shortest and tallest edge heights)

Worked Example

Problem: A circular cylinder with radius 5 cm is truncated by an oblique plane. The shortest height along the side is 8 cm and the tallest height is 12 cm. Find the volume of the truncated cylinder.

Step 1: Find the area of the circular base.

A_{\text{base}} = \pi r^2 = \pi (5)^2 = 25\pi \text{ cm}^2

Step 2: Calculate the average height. For a cylinder cut by a plane, the average height is the mean of the minimum and maximum heights.

h_{\text{avg}} = \frac{h_{\min} + h_{\max}}{2} = \frac{8 + 12}{2} = 10 \text{ cm}

Step 3: Multiply the base area by the average height to get the volume.

V = A_{\text{base}} \times h_{\text{avg}} = 25\pi \times 10 = 250\pi \approx 785.4 \text{ cm}^3

Answer: The volume of the truncated cylinder is

250\pi \approx 785.4

cm³.

Another Example

This example uses a triangular prism instead of a cylinder. With a prism, you average the heights at each vertex of the polygonal base rather than using just the min and max heights.

Problem: A triangular prism has a base that is an equilateral triangle with side length 6 cm. The prism is truncated by an oblique plane so that the three lateral edge heights measured from the base to the cutting plane are 9 cm, 12 cm, and 15 cm. Find the volume of the truncated prism.

Step 1: Find the area of the equilateral triangular base.

A_{\text{base}} = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4}(6)^2 = 9\sqrt{3} \approx 15.59 \text{ cm}^2

Step 2: For a prism, the average height is the arithmetic mean of all the lateral edge heights (one per vertex of the base).

h_{\text{avg}} = \frac{h_1 + h_2 + h_3}{3} = \frac{9 + 12 + 15}{3} = 12 \text{ cm}

Step 3: Compute the volume using the base area and average height.

V = A_{\text{base}} \times h_{\text{avg}} = 9\sqrt{3} \times 12 = 108\sqrt{3} \approx 187.1 \text{ cm}^3

Answer: The volume of the truncated triangular prism is

108\sqrt{3} \approx 187.1

cm³.

Frequently Asked Questions

What is the difference between a truncated cylinder and a truncated cone?

A truncated cylinder (or prism) keeps the same cross-sectional shape throughout—only one base is sliced off by a plane, while the lateral edges remain parallel. A truncated cone (or pyramid), also called a frustum, has both bases intact but they are different sizes because the solid tapers. In a frustum, two parallel cuts produce two similar bases; in a truncated cylinder, one base is the original and the other is the angled cut.

Why does the average-height formula work for a truncated cylinder?

When a plane cuts through a right cylinder at an angle, the resulting solid can be thought of as the original full cylinder plus or minus a wedge-shaped piece. By symmetry, the wedge has zero net contribution when you average the heights across the base. This makes the volume equal to the base area times the height at the centroid of the base, which equals the average of the minimum and maximum heights for a circular cross-section.

How do you find the surface area of a truncated cylinder?

The total surface area includes three parts: the original circular base (

\pi r^2

), the elliptical top created by the oblique cut (

\pi r^2 / \cos\theta

, where

\theta

is the angle of the cutting plane relative to the base), and the lateral surface area. The lateral area equals the circumference times the average height:

2\pi r \cdot h_{\text{avg}}

. Add all three to get the total.

Truncated Cylinder/Prism vs. Truncated Cone/Pyramid (Frustum)

	Truncated Cylinder/Prism	Truncated Cone/Pyramid (Frustum)
What is cut	One base is sliced off by an oblique or parallel plane	The apex region is removed by a plane parallel to the base
Number of intact original bases	One (the uncut base)	One (the original base); the top is a new, smaller similar shape
Lateral edges	Remain parallel to each other	Converge toward where the apex would have been
Volume formula	$V = A_{\text{base}} \times h_{\text{avg}}$	$V = \frac{h}{3}(A_1 + A_2 + \sqrt{A_1 A_2})$
Cross-sections parallel to base	All identical (same shape and size)	Similar shapes that change in size

Why It Matters

Truncated cylinders and prisms appear frequently in engineering and architecture—think of pipes cut at an angle to join at a junction, or columns sliced to meet a sloped roof. In calculus, computing the volume of a truncated cylinder is a standard application of integration, and the average-height formula provides a useful shortcut. Understanding truncation also builds intuition for how cross-sectional methods work in finding volumes of irregular solids.

Common Mistakes

Mistake: Confusing a truncated cylinder with a frustum (truncated cone). Students sometimes apply the frustum formula to a truncated cylinder.

Correction: A truncated cylinder has parallel lateral edges and a uniform cross-section. Its volume is simply base area times average height. The frustum formula applies only when the solid tapers (as in a cone or pyramid).

Mistake: For a truncated prism, averaging only two heights instead of all vertex heights.

Correction: You must average the heights at every vertex of the polygonal base. For a triangular base, average three heights; for a rectangular base, average four heights. Using only the min and max gives the correct result for a cylinder (by symmetry) but not for a general prism.

Related Terms

Cylinder — The parent solid before truncation
Prism — The parent solid before truncation
Base — The intact face of the truncated solid
Plane — The cutting surface that truncates the solid
Oblique — Describes a non-perpendicular cutting angle
Parallel Planes — Special case where cut is parallel to the base
Truncated Cone or Pyramid — Related solid where the apex is removed instead