100 LearnersLast updated on July 17th, 2025
Volume of Square Pyramid
The volume of a square pyramid is the total space it occupies or the number of cubic units it can hold. A square pyramid is a 3D shape with a square base and four triangular faces that meet at a single point called the apex. To find the volume of a square pyramid, we use its base area and height. In real life, kids relate to the volume of a square pyramid by thinking of things like the pyramids of Egypt or a party hat. In this topic, let’s learn about the volume of a square pyramid.
What is the volume of a square pyramid?
General Formula:
Volume = (1/3) × a² × h
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Base Area: Area of the square base.
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Height (h): Perpendicular distance from the base to the apex.
Specific Formula for Square Base
Volume = (1/3) × a² × h
Where:
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a = side length of the square base
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h = height of the pyramid
How to Derive the Volume of a Square Pyramid?
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General Formula for Volume of a Square Pyramid:
Volume = (1/3) × Base Area × Height -
With Base Area as a² (for square base):
Volume = (1/3) × a² × h
Where:
-
a= side length of the square base -
h= vertical height of the pyramid
How to find the volume of a square pyramid?
The volume of a square pyramid is always expressed in cubic units, for example, cubic centimeters (cm³), cubic meters (m³). Calculate the base area, multiply it by the height, and then divide by three to find the volume.
Let’s take a look at the formula for finding the volume of a square pyramid: Write down the formula: \
Volume = (1/3) × a² × h
The base area is \(a2\) where \(a\) is the side length of the square base.
Once we know the base area and height, substitute those values into the formula:
Volume = (1/3) × a² × h
Tips and Tricks for Calculating the Volume of Square Pyramid
Remember the formula: The formula for the volume of a square pyramid is straightforward: \Volume = (1/3) × a² × h
Break it down: The volume is how much space fits inside the pyramid.
Calculate the base area and multiply by the height, then divide by three. Simplify the numbers: If the base side length is a simple number like 2, 3, or 4, it is easy to square and multiply.
For example, if \(a = 3\), then \(a2 = 9\). Check for any measurement errors: Ensure all measurements are in the same unit before calculating the volume.
Common Mistakes and How to Avoid Them in Volume of Square Pyramid
Making mistakes while learning the volume of the square pyramid is common. Let’s look at some common mistakes and how to avoid them to get a better understanding of the volume of square pyramids.
Mistake 1
Confusing Volume with Surface Area
Some students confuse the formula for volume with the formula for surface area.
Surface area involves all the triangular faces and the base, while volume is calculated using the base area and height.
For example, the volume is \Volume = (1/3) × a² × h
not the sum of the areas of the faces.
Mistake 2
Confusing Volume with Perimeter
Some kids may think of the pyramid's perimeter instead of the volume formula.
Volume is the space inside the pyramid, whereas perimeter refers to the total length around the base of the 2D shape.
Do not mix them up.
Mistake 3
Using the wrong Formula for different pyramids
Some kids use the formula for the volume of a different type of pyramid or a prism instead of the square pyramid formula.
Mistake 4
Confusing square base area with linear measurement
Thinking of the base area in terms of linear measurements.
This happens when someone uses the side length \(a\) without squaring it to find the base area.
Mistake 5
Incorrectly calculating the height
Some students calculate the given volume without correctly solving for the perpendicular height. Ensure the height used is perpendicular to the base.
Volume of Square Pyramid Examples
Problem 1
A square pyramid has a base side length of 4 cm and a height of 9 cm. What is its volume?
The volume of the square pyramid is 48 cm³.
Explanation
To find the volume of a square pyramid, use the formula: Volume = (1/3) × a² × h
Here, the base side length \(a\) is 4 cm, and the height \(h\) is 9 cm, so:
V = (1/3) × 4² × 9 = (1/3) × 16 × 9 = 48 cm³
Problem 2
A square pyramid has a base side length of 6 m and a height of 12 m. Find its volume.
The volume of the square pyramid is 144 m³.
Explanation
To find the volume of a square pyramid, use the formula: Volume = (1/3) × a² × h
Substitute the base side length (6 m) and height (12 m):
V = (1/3) × 6² × 12 = (1/3) × 36 × 12 = 144 m³
Problem 3
The volume of a square pyramid is 75 cm³. If its base side length is 5 cm, what is its height?
The height of the square pyramid is 9 cm.
Explanation
To find the height when the volume is known, rearrange the volume formula:
Volume = (1/3) × a² × h
Substitute the volume (75 cm³) and base side length (5 cm):
Height = (3 × 75) / (5²) = 225 / 25 = 9 cm
Problem 4
A square pyramid has a base side length of 3 inches and a height of 8 inches. Find its volume.
The volume of the square pyramid is 24 inches³.
Explanation
Using the formula for volume: Volume = (1/3) × a² × h
Substitute the base side length (3 inches) and height (8 inches): \
V = (1/3) × a² × h
V = (1/3) × 3² × 8
V = (1/3) × 9 × 8
V = 24 inches³
Problem 5
You have a square pyramid-shaped tent with a base side length of 5 feet and a height of 10 feet. How much space (in cubic feet) is available inside the tent?
The tent has a volume of 83.33 cubic feet.
Explanation
Using the formula for volume:Volume = (1/3) × a² × h
Substitute the base side length (5 feet) and height (10 feet):
V = (1/3) × 5² × 10
V = (1/3) × 25 × 10
V = 83.33 ft³
FAQs on Volume of Square Pyramid
1.Is the volume of a square pyramid the same as the surface area?
2.How do you find the volume if the base side length and height are given?
3.What if I have the volume and need to find the height?
4.Can the base side length be a decimal or fraction?
5.Is the volume of a square pyramid the same as that of other pyramids?
Important Glossaries for Volume of Square Pyramid
- Base: The square bottom face of the pyramid, where 'base area' is calculated as \(a2\).
- Height: The perpendicular distance from the base to the apex of the pyramid.
- Volume: The amount of space enclosed within a 3D object, like a square pyramid, calculated using base area and height.
- Apex: The topmost point where the triangular faces of the pyramid converge.
- Square Pyramid: A pyramid with a square base and four triangular faces meeting at the apex.
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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
