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144 LearnersLast updated on December 11, 2025
Volume of Truncated Cone
The volume of a truncated cone is the total space it occupies. A truncated cone, also known as a frustum of a cone, is formed by slicing the top off a cone parallel to its base, resulting in two circular ends with different radii. To find the volume of a truncated cone, we use the formula involving the radii of the two circular ends and the height. In real life, examples of truncated cones include lampshades and buckets. In this topic, let’s learn about the volume of a truncated cone.
What is the Volume of a Truncated Cone?
The volume of a truncated cone is the amount of space it occupies.
It is calculated using the formula: Volume = (1/3)πh(R² + r² + Rr) Where ‘R’ is the radius of the larger base, ‘r’ is the radius of the smaller base, and ‘h’ is the height of the truncated cone.
A truncated cone is a 3-dimensional shape with two circular ends of different sizes. To calculate its volume, we add the areas of the two circular ends and their geometric mean, and then multiply by the height and π divided by 3.
The formula for the volume of a truncated cone is given as follows: Volume = (1/3)πh(R² + r² + Rr)
How to Derive the Volume of a Truncated Cone?
To derive the volume of a truncated cone, we start with the concept of volume for a cone.
A truncated cone is formed by cutting a smaller cone off the top of a larger cone.
The volume of a cone is: Volume = (1/3)πr²h
For a truncated cone: Volume = (1/3)πh(R² + r² + Rr)
This formula is derived by subtracting the volume of the smaller cone from that of the larger cone.
How to Find the Volume of a Truncated Cone?
The volume of a truncated cone is expressed in cubic units, for example, cubic centimeters (cm³) or cubic meters (m³).
To find the volume, follow these steps: Write down the formula: Volume = (1/3)πh(R² + r² + Rr) Where R is the radius of the larger base, r is the radius of the smaller base, and h is the height.
Substitute the values of R, r, and h into the formula to find the volume. Ensure all measurements are in the same units.
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Tips and Tricks for Calculating the Volume of a Truncated Cone
Remember the formula: Volume = (1/3)πh(R² + r² + Rr) Break it down: Understanding the formula helps in visualizing the problem. The volume is the space inside the truncated cone.
Simplify the numbers: For ease of calculation, ensure the units are consistent, and simplify the expression (R² + r² + Rr) before multiplying by πh/3.
Check for errors in measurement: Ensure you have the correct radii and height, as mistakes in these values will lead to incorrect volume.
Common Mistakes and How to Avoid Them in Volume of Truncated Cone
Learning the volume of a truncated cone can lead to some common mistakes. Let’s look at these mistakes and learn how to avoid them.
Mistake 1
Confusing the Formula with a Complete Cone
Some students mistakenly use the formula for a complete cone: Volume = (1/3)πr²h, instead of the truncated cone formula: Volume = (1/3)πh(R² + r² + Rr).
Mistake 2
Using Incorrect Radii
Ensure that R and r are correctly identified as the radii of the larger and smaller bases, respectively. Using the wrong values will result in an incorrect volume calculation.
Mistake 3
Neglecting Height in the Formula
Some students forget to include the height h in the formula. Remember, the height is crucial for calculating the volume.
Mistake 4
Incorrect Unit Conversion
Be consistent with units. If the radii are in centimeters and the height in meters, convert them to the same unit before substituting into the formula.
Mistake 5
Misunderstanding the Formula
Misinterpreting the formula as being only for circular areas and not understanding it involves a combination of the areas of the two ends and their geometric mean.
Volume of Truncated Cone Examples
Problem 1
A truncated cone has a height of 5 cm, a larger base radius of 4 cm, and a smaller base radius of 2 cm. What is its volume?
The volume of the truncated cone is approximately 125.67 cm³.
Explanation
To find the volume of a truncated cone, use the formula: V = (1/3)πh(R² + r² + Rr)
Here, R = 4 cm, r = 2 cm, and h = 5 cm,
so: V = (1/3)π * 5 * (4² + 2² + 4*2) V = (1/3)π * 5 * (16 + 4 + 8) V = (1/3)π * 5 * 28 V ≈ 125.67 cm³
Problem 2
A truncated cone has a height of 10 m, a larger base radius of 6 m, and a smaller base radius of 3 m. Find its volume.
The volume of the truncated cone is approximately 942.48 m³.
Explanation
To find the volume of a truncated cone, use the formula: V = (1/3)πh(R² + r² + Rr)
Substitute the values: R = 6 m, r = 3 m, h = 10 m:
V = (1/3)π * 10 * (6² + 3² + 6*3) V = (1/3)π * 10 * (36 + 9 + 18) V = (1/3)π * 10 * 63 V ≈ 942.48 m³
Problem 3
The volume of a truncated cone is 250 cm³. If the height is 5 cm and the larger base radius is 5 cm, what is the smaller base radius?
The smaller base radius of the truncated cone is approximately 2.29 cm.
Explanation
Using the formula for volume: V = (1/3)πh(R² + r² + Rr)
Rearrange to solve for r: 250 = (1/3)π * 5 * (5² + r² + 5r) 250 = (5/3)π * (25 + r² + 5r)
Rearrange and solve for r using algebraic methods, yielding r ≈ 2.29 cm.
Problem 4
A truncated cone has a height of 7 inches, a larger base radius of 8 inches, and a smaller base radius of 3 inches. Find its volume.
The volume of the truncated cone is approximately 861.9 inches³.
Explanation
Using the formula for volume: V = (1/3)πh(R² + r² + Rr)
Substitute the values: R = 8 inches, r = 3 inches, h = 7 inches:
V = (1/3)π * 7 * (8² + 3² + 8*3)
V = (1/3)π * 7 * (64 + 9 + 24)
V = (1/3)π * 7 * 97
V ≈ 861.9 inches³
Problem 5
You have a lampshade shaped like a truncated cone with a height of 12 cm, a larger base radius of 10 cm, and a smaller base radius of 5 cm. How much space (in cubic centimeters) does it occupy?
The lampshade occupies approximately 2,617.99 cm³.
Explanation
Using the formula for volume: V = (1/3)πh(R² + r² + Rr)
Substitute the values: R = 10 cm, r = 5 cm, h = 12 cm:
V = (1/3)π * 12 * (10² + 5² + 10*5)
V = (1/3)π * 12 * (100 + 25 + 50)
V = (1/3)π * 12 * 175 V ≈ 2,617.99 cm³
FAQs on Volume of Truncated Cone
1.Is the volume of a truncated cone the same as a complete cone?
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 or height be decimals or fractions?
5.Is the volume of a truncated cone the same as the surface area?
Important Glossaries for Volume of Truncated Cone
- Radius (R, r): The distance from the center to the edge of the base circles of the truncated cone. R is the radius of the larger base, and r is the radius of the smaller base.
- Height (h): The perpendicular distance between the two bases of the truncated cone.
- Volume: The amount of space enclosed within a 3D object. For a truncated cone, it is given by the formula V = (1/3)πh(R² + r² + Rr).
- Cubic Units: The units of measurement used for volume. If the measurements are in centimeters, the volume will be in cubic centimeters (cm³).
- Truncated Cone: A 3D geometric shape formed by slicing the top off a cone parallel to its base, resulting in two circular ends with different radii.
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
