102 LearnersLast updated on July 20th, 2025
Volume of Frustum
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.
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.
Mistake 1
Confusing Volume with Surface Area
Some students confuse the formula for volume with the formula for surface area.
Surface area involves calculating the area of the lateral and base surfaces, whereas the volume uses the radii and height.
Remember, volume for frustum is calculated using the formula: Volume = (1/3) * π * h * (R² + r² + R * r)
Mistake 2
Confusing Volume with Perimeter
Some kids may think of the perimeter of the bases instead of the volume formula.
Volume is the space inside the frustum, whereas perimeter refers to the total length around the edges of a 2D shape. Do not mix them up.
Mistake 3
Using the wrong Formula for other shapes
Some kids use the formula for the volume of a cylinder or cone instead of the frustum formula. Make sure to use: Volume = (1/3) * π * h * (R² + r² + R * r)
Mistake 4
Confusing cubic volume with linear volume
Thinking of volume in terms of linear measurements.
This happens when someone uses linear dimensions without cubing them. Volume relates to cubic measurements.
Mistake 5
Incorrectly calculating the radii or height
Some students may incorrectly measure or calculate the given dimensions, leading to errors in finding the volume. Double-check all measurements before calculation.
Volume of Frustum Examples
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?
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
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.
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
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?
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.
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.
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
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?
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
FAQs on Volume of Frustum
1.Is the volume of a frustum the same as the surface area?
2.How do you find the volume if the radii and height are given?
3.What if I have the volume and need to find one of the radii?
4.Can the radii be decimals or fractions?
5.Is the volume of a frustum the same as the surface area?
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
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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.
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