Volume of Frustum of Pyramid
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Last updated on September 5, 2025

The volume of a frustum of a pyramid is the total space it occupies or the number of cubic units it can hold. A frustum of a pyramid is formed when a pyramid is cut parallel to its base, creating a smaller top base and a larger bottom base. To find the volume of a frustum of a pyramid, we use a specific formula that involves the areas of the two bases and the height of the frustum. In this topic, let’s learn about the volume of the frustum of a pyramid.

What is the Volume of the Frustum of a Pyramid?

The volume of a frustum of a pyramid is the amount of space it occupies. It is calculated using the formula:  Volume = 1/3 x h x (A_1 + A_2 + √A_1 x A_2 Where  h  is the height of the frustum,  A_1  is the area of the larger base, and  A_2 is the area of the smaller base.

How to Derive the Volume of a Frustum of a Pyramid?

To derive the volume of a frustum of a pyramid, we consider the volume as the total space occupied by the 3D object. The formula is derived by subtracting the volume of the smaller pyramid (at the top)

from the volume of the larger pyramid (original pyramid): Volume = 1/3 x h x (A_1 + A_2 + √{A_1 \times A_2})  This accounts for the areas of both bases and the height of the frustum.

How to Find the Volume of a Frustum of a Pyramid?

The volume of a frustum of a pyramid is expressed in cubic units, such as cubic centimeters (cm³) or cubic meters (m³).

To find the volume, use the formula: Volume = 1/3 x h x (A_1 + A_2 + √{A_1 \times A_2})  Where  h  is the height,  A_1  is the area of the larger base, and  A_2  is the area of the smaller base. Substitute these values into the formula to find the volume.

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Tips and Tricks for Calculating the Volume of a Frustum of a Pyramid

Remember the formula: The formula for the volume of a frustum of a pyramid is: Volume = 1/3 x h x(A_1 + A_2 + √{A_1 \times A_2})

Break it down: Understand each part of the formula. Calculate the areas of both bases first, then use these in the formula.

Simplify calculations: Work step-by-step by calculating each part of the formula separately before combining them.

Unit consistency: Ensure all measurements are in the same unit before calculating the volume.

Common Mistakes and How to Avoid Them in Volume of Frustum of Pyramid

Making mistakes while learning the volume of a frustum of a pyramid is common. Let’s look at some common mistakes and how to avoid them to get a better understanding.

Problem 1

A frustum of a pyramid has a height of 6 cm, a larger base area of 25 cm², and a smaller base area of 9 cm². What is its volume?

Okay, lets begin

The volume of the frustum of the pyramid is 124 cm³.

Explanation

Using the formula for volume: Volume = 1/3 x 6 x (25 + 9 + √{25 x 9})   = 2 x (34 + 15)  = 2 x 49 = 98 cm3 

Well explained 👍

Problem 2

A frustum of a pyramid has a height of 10 m, a larger base area of 100 m², and a smaller base area of 40 m². Find its volume.

Okay, lets begin

The volume of the frustum of the pyramid is 1464.1 m³.

Explanation

Using the formula for volume: Volume = 1/3 x 10 x (100 + 40 +√{100 x 40})  = 10/3 x (140 + 63.25)  = 10/3 x 203.25 approx 677.5 m3 

Well explained 👍

Problem 3

The volume of a frustum of a pyramid is 500 cm³. If the height is 8 cm and the larger base area is 30 cm², what is the area of the smaller base?

Okay, lets begin

The area of the smaller base is approximately 18.75 cm².

Explanation

Using the formula for volume:  500 = 1/3 x 8 x (30 + A_2 + √{30 x A_2}) Solving for A_2  involves algebraic manipulation, and the exact calculation will depend on solving this equation.

Well explained 👍

Problem 4

A frustum of a pyramid has a height of 5 inches, a larger base area of 50 in², and a smaller base area of 20 in². Find its volume.

Okay, lets begin

The volume of the frustum of the pyramid is 257.08 inches³.

Explanation

Using the formula for volume: Volume = 1/3 x 5 x (50 + 20 + √{50 x 20})  = 5/3 x (70 + 31.62)  = 5/3 x101.62 approx 169.37 in3 

Well explained 👍

Problem 5

You have a frustum of a pyramid with a height of 7 feet, a larger base area of 60 ft², and a smaller base area of 15 ft². How much space (in cubic feet) does it occupy?

Okay, lets begin

The frustum of the pyramid occupies approximately 406.42 cubic feet.

Explanation

Using the formula for volume: Volume = 1/3 x 7 x (60 + 15 + √{60 x 15}) = 7/3 x(75 + 30) = 7/3 x 105 approx 245 ft3

Well explained 👍

FAQs on Volume of Frustum of Pyramid

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

No, the volume and surface area of a frustum of a pyramid are different concepts: Volume refers to the space inside the frustum and is given by the formula involving the areas of the bases and height. Surface area refers to the total area of all the faces and bases of the frustum.

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

To calculate the volume when the base areas and height are provided, use the formula: \[ \text{Volume} = \frac{1}{3} \times h \times (A_1 + A_2 + \sqrt{A_1 \times A_2}) \]

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

If the volume is given and you need to find the area of one base, rearrange the volume formula to solve for the unknown base area. This may involve algebraic manipulation.

4.Can the base areas be decimals or fractions?

Yes, the base areas of a frustum of a pyramid can be decimals or fractions. The formula can accommodate these values to calculate the volume accurately.

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

No, the volume and surface area of a frustum of a pyramid are different concepts: volume refers to the space inside the frustum and is calculated using the formula involving the base areas and height.

Important Glossaries for Volume of Frustum of Pyramid

  • Height: The perpendicular distance between the two bases of the frustum.
  • Larger Base Area ( A_1 ): The area of the larger base of the frustum.
  • Smaller Base Area ( A_2 ): The area of the smaller base of the frustum.
  • Volume: The amount of space enclosed within the frustum, calculated using the formula.
  • Cubic Units: The units of measurement used for volume; e.g., cubic centimeters (cm³), cubic meters (m³).

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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