Truncated Cone or Pyramid

Q: How do you find the volume of a truncated cone?

For a frustum (parallel cut), use $V = \frac{h}{3}(A_1 + A_2 + \sqrt{A_1 \cdot A_2})$, where $h$ is the perpendicular height and $A_1$, $A_2$ are the two base areas. For a cone with circular bases, this simplifies to $V = \frac{\pi h}{3}(r_1^2 + r_2^2 + r_1 r_2)$. If the cut is oblique (not parallel), the calculation is more complex and typically requires integration.

Truncated Cone or Pyramid

A cone or pyramid which has its apex cut off by an intersecting plane. The plane may be either oblique or parallel to the base.

Note: If the truncating plane is parallel to the base the figure is called a frustum.

3D truncated cone with a small circular top face, wider circular base, and slanted sides, labeled "Truncated Cone

See also

Truncated cylinder or prism

Key Formula

V = \frac{h}{3}\left(A_1 + A_2 + \sqrt{A_1 \cdot A_2}\right)

Where:

$V$ = Volume of the frustum (truncated solid with parallel cut)
$h$ = Perpendicular height between the two parallel bases
$A_1$ = Area of the larger base (original base)
$A_2$ = Area of the smaller base (created by the cut)

Worked Example

Problem: A cone with a base radius of 6 cm is cut by a plane parallel to the base, creating a frustum (truncated cone). The smaller top circle has a radius of 3 cm, and the perpendicular height of the frustum is 8 cm. Find the volume.

Step 1: Calculate the area of the larger base using the circle area formula.

A_1 = \pi r_1^2 = \pi(6)^2 = 36\pi \text{ cm}^2

Step 2: Calculate the area of the smaller base.

A_2 = \pi r_2^2 = \pi(3)^2 = 9\pi \text{ cm}^2

Step 3: Find the geometric mean of the two base areas.

\sqrt{A_1 \cdot A_2} = \sqrt{36\pi \cdot 9\pi} = \sqrt{324\pi^2} = 18\pi \text{ cm}^2

Step 4: Substitute all values into the frustum volume formula.

V = \frac{8}{3}(36\pi + 9\pi + 18\pi) = \frac{8}{3}(63\pi)

Step 5: Simplify to get the final volume.

V = \frac{504\pi}{3} = 168\pi \approx 527.8 \text{ cm}^3

Answer: The volume of the truncated cone is

168\pi \approx 527.8

cm³.

Another Example

This example uses a truncated pyramid with square bases rather than a truncated cone with circular bases, showing that the same volume formula works for both shapes.

Problem: A square pyramid with a base side length of 10 m is cut parallel to its base to form a frustum. The top square has a side length of 4 m, and the height of the frustum is 9 m. Find its volume.

Step 1: Calculate the area of the larger square base.

A_1 = 10^2 = 100 \text{ m}^2

Step 2: Calculate the area of the smaller square base.

A_2 = 4^2 = 16 \text{ m}^2

Step 3: Find the geometric mean of the two base areas.

\sqrt{A_1 \cdot A_2} = \sqrt{100 \cdot 16} = \sqrt{1600} = 40 \text{ m}^2

Step 4: Apply the frustum volume formula.

V = \frac{9}{3}(100 + 16 + 40) = 3 \times 156 = 468 \text{ m}^3

Answer: The volume of the truncated pyramid is 468 m³.

Frequently Asked Questions

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

A frustum is a specific type of truncated cone (or pyramid) where the cutting plane is parallel to the base, so the top and bottom faces are similar shapes. A truncated cone is a more general term — the cutting plane can be oblique (tilted at an angle), producing a top face that is not parallel to the base. Every frustum is a truncated cone, but not every truncated cone is a frustum.

How do you find the volume of a truncated cone?

For a frustum (parallel cut), use

V = \frac{h}{3}(A_1 + A_2 + \sqrt{A_1 \cdot A_2})

, where

h

is the perpendicular height and

A_1

A_2

are the two base areas. For a cone with circular bases, this simplifies to

V = \frac{\pi h}{3}(r_1^2 + r_2^2 + r_1 r_2)

. If the cut is oblique (not parallel), the calculation is more complex and typically requires integration.

What does truncated mean in geometry?

Truncated means that a portion of a solid has been cut off by a plane. The term comes from the Latin word 'truncare,' meaning to cut short. You can truncate cones, pyramids, cylinders, prisms, and even polyhedra like cubes and icosahedra.

Truncated Cone/Pyramid vs. Frustum

	Truncated Cone/Pyramid	Frustum
Definition	A cone or pyramid with its apex removed by any intersecting plane	A cone or pyramid with its apex removed by a plane parallel to the base
Cutting plane	Can be oblique or parallel to the base	Must be parallel to the base
Top face shape	May or may not be similar to the base	Always similar to the base
Volume formula	Standard formula works only for parallel cut; oblique requires integration	V = (h/3)(A₁ + A₂ + √(A₁·A₂)) always applies
Relationship	General category	Special case of a truncated solid

Why It Matters

Truncated cones and pyramids appear frequently in engineering and architecture — think of lampshades, buckets, dam walls, and ancient structures like ziggurats. In geometry courses, frustum volume problems are common on tests and require you to combine your knowledge of base areas, heights, and the prismoidal formula. Understanding the distinction between a general truncation and a frustum also deepens your grasp of how cross-sections relate to three-dimensional solids.

Common Mistakes

Mistake: Using the formula V = (h/3)(A₁ + A₂ + √(A₁·A₂)) when the cut is oblique (not parallel to the base).

Correction: This formula only applies when the two bases are parallel (i.e., the solid is a frustum). An oblique truncation produces a non-standard shape that requires different methods, such as calculus-based integration, to find the volume.

Mistake: Averaging the two base areas and multiplying by the height, treating the frustum like a prism.

Correction: The formula V = h × (A₁ + A₂)/2 overestimates the volume. You must include the geometric mean term √(A₁·A₂) and divide by 3. The correct formula is V = (h/3)(A₁ + A₂ + √(A₁·A₂)).

Related Terms

Cone — The original solid before truncation
Pyramid — The original solid before truncation
Frustum of a Cone or Pyramid — Special case with a parallel cut
Apex — The point removed by truncation
Plane — The flat surface that makes the cut
Parallel Planes — Defines when a frustum is formed
Base — Both the original and new face created
Truncated Cylinder or Prism — Truncation applied to non-tapered solids