Volume of Frustum
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Last updated on August 5, 2025

The volume of a frustum is the total space it occupies or the number of cubic units it can hold. A frustum is a 3D shape typically created by slicing the top off a cone or pyramid parallel to its base. To find the volume of a frustum, we use a specific formula that involves the radii of the top and bottom bases, as well as the height. In real life, kids can relate to the volume of a frustum by thinking of objects like a lampshade or a bucket. In this topic, let’s learn about the volume of a frustum.

What is the volume of a frustum?

The volume of a frustum is the amount of space it occupies.

It is calculated by using the formula: Volume = (1/3) * π * h * (R² + r² + R * r) Where R is the radius of the bottom base, r is the radius of the top base, and h is the height of the frustum.

Volume of Frustum Formula A frustum is a 3-dimensional shape formed by slicing a cone or pyramid.

To calculate its volume, we use the radii of the top and bottom bases along with the height.

The formula for the volume of a frustum is given as follows: Volume = (1/3) * π * h * (R² + r² + R * r)

How to Derive the Volume of a Frustum?

To derive the volume of a frustum, we use the concept of volume as the total space occupied by a 3D object.

The frustum can be seen as the difference in volume between two cones or pyramids with a common axis.

The formula for the volume of a frustum is derived by subtracting the volume of the smaller cone (or pyramid) from the larger one: Volume of Cone = (1/3) * π * h * R² Volume of smaller Cone = (1/3) * π * (h - h₁) * r² Where h is the height of the larger cone and h₁ is the height of the smaller cone.

The volume of the frustum will be: Volume = Volume of larger Cone - Volume of smaller Cone Volume = (1/3) * π * h * (R² + r² + R * r)

How to find the volume of a frustum?

The volume of a frustum is always expressed in cubic units, for example, cubic centimeters (cm³), cubic meters (m³).

Use the formula involving the radii of the bases and height, to find the volume. Let’s take a look at the formula for finding the volume of a frustum: Write down the formula Volume = (1/3) * π * h * (R² + r² + R * r)

The radii R and r are the radii of the bottom and top bases.

The height h is the perpendicular distance between the bases. Once we know the values of R, r, and h, substitute them into the formula to find the volume of the frustum.

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Tips and Tricks for Calculating the Volume of Frustum

Remember the formula: The formula for the volume of a frustum is: Volume = (1/3) * π * h * (R² + r² + R * r)

Break it down: The volume is how much space fits inside the frustum. Use the values of radii and height carefully.

Simplify the numbers: If the radii and height are simple numbers, simplify calculations by breaking down the components, for example, R², r², and R * r.

Check for accuracy: Ensure you have the correct measurements for the radii and height before substituting them into the formula.

Common Mistakes and How to Avoid Them in Volume of Frustum

Making mistakes while learning the volume of the frustum is common.

Let’s look at some common mistakes and how to avoid them to get a better understanding of the volume of frustums.

Problem 1

A frustum of a cone has a height of 5 cm, a bottom radius of 3 cm, and a top radius of 2 cm. What is its volume?

Okay, lets begin

The volume of the frustum is approximately 83.78 cm³.

Explanation

To find the volume of a frustum, use the formula: V = (1/3) * π * h * (R² + r² + R * r) Here, R = 3 cm, r = 2 cm, and h = 5 cm, so: V ≈ (1/3) * π * 5 * (3² + 2² + 3 * 2) V ≈ (1/3) * π * 5 * (9 + 4 + 6) V ≈ (1/3) * π * 5 * 19 V ≈ 83.78 cm³

Well explained 👍

Problem 2

A frustum of a pyramid has a height of 8 m, a bottom square base with a side of 6 m, and a top square base with a side of 4 m. Find its volume.

Okay, lets begin

The volume of the frustum is approximately 320 m³.

Explanation

To find the volume of a frustum of a pyramid, use: V = (1/3) * h * (A₁ + A₂ + √(A₁A₂)) Where A₁ = 6² = 36 m², A₂ = 4² = 16 m², and h = 8 m. V ≈ (1/3) * 8 * (36 + 16 + √(36 * 16)) V ≈ (1/3) * 8 * (36 + 16 + 24) V ≈ (1/3) * 8 * 76 V ≈ 320 m³

Well explained 👍

Problem 3

The volume of a frustum of a cone is 150 cm³, with a height of 10 cm and a bottom radius of 5 cm. What is the top radius?

Okay, lets begin

The top radius of the cone is approximately 2.22 cm.

Explanation

Using the volume formula, solve for r: 150 = (1/3) * π * 10 * (5² + r² + 5 * r) 450 = π * (25 + r² + 5r) 450/π = 25 + r² + 5r Solve the quadratic equation for r.

Well explained 👍

Problem 4

A frustum of a cone has a height of 4 inches, a bottom radius of 7 inches, and a top radius of 3 inches. Find its volume.

Okay, lets begin

The volume of the frustum is approximately 527.79 inches³.

Explanation

Using the formula for volume: V = (1/3) * π * h * (R² + r² + R * r)

Substitute the values R = 7 inches, r = 3 inches, h = 4 inches: V ≈ (1/3) * π * 4 * (7² + 3² + 7 * 3) V ≈ (1/3) * π * 4 * (49 + 9 + 21) V ≈ (1/3) * π * 4 * 79 V ≈ 527.79 inches³

Well explained 👍

Problem 5

You have a frustum-shaped bucket with a height of 12 feet, a bottom radius of 5 feet, and a top radius of 3 feet. How much space (in cubic feet) is available inside the bucket?

Okay, lets begin

The bucket has a volume of approximately 1,256.64 cubic feet.

Explanation

Using the formula for volume: V = (1/3) * π * h * (R² + r² + R * r)

Substitute the values R = 5 feet, r = 3 feet, h = 12 feet: V ≈ (1/3) * π * 12 * (5² + 3² + 5 * 3) V ≈ (1/3) * π * 12 * (25 + 9 + 15) V ≈ (1/3) * π * 12 * 49 V ≈ 1,256.64 ft³

Well explained 👍

FAQs on Volume of Frustum

1.Is the volume of a frustum the same as the surface area?

No, the volume and surface area of a frustum are different concepts.

Volume refers to the space inside the frustum and is given by V = (1/3) * π * h * (R² + r² + R * r), while surface area involves calculating the lateral and base areas.

2.How do you find the volume if the radii and height are given?

To calculate the volume when the radii and height are provided, use the formula: Volume = (1/3) * π * h * (R² + r² + R * r). Substitute the given values into the formula.

3.What if I have the volume and need to find one of the radii?

If the volume of the frustum is given and you need to find one of the radii, rearrange the volume formula to solve for the unknown radius.

This may involve solving a quadratic equation.

4.Can the radii be decimals or fractions?

Yes, the radii of a frustum can be decimals or fractions. The volume calculation remains the same: use the formula with the given values.

5.Is the volume of a frustum the same as the surface area?

No, the volume and surface area of a frustum are different concepts.

Volume refers to the space inside the frustum, while surface area involves calculating the lateral and base areas.

Important Glossaries for Volume of Frustum

  • Radii: The radii of the top and bottom bases of the frustum.
  • Height: The perpendicular distance between the two bases of the frustum.
  • Frustum: A 3D shape formed by slicing the top off a cone or pyramid parallel to its base.
  • Volume: The amount of space enclosed within a 3D object, calculated for a frustum using the formula: V = (1/3) * π * h * (R² + r² + R * r).
  • Cubic Units: The units of measurement used for volume, expressed as cm³, m³, etc.

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Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

Fun Fact

: She has songs for each table which helps her to remember the tables