105 LearnersLast updated on July 20th, 2025
Volume of Tetrahedron
The volume of a tetrahedron is the total space it occupies or the number of cubic units it can hold. A tetrahedron is a 3D shape with four triangular faces. To find the volume of a regular tetrahedron, we use the formula involving the side length. In real life, the volume of a tetrahedron can be related to things like pyramids or certain types of dice. In this topic, let’s learn about the volume of a tetrahedron.
What is the volume of a tetrahedron?
The volume of a tetrahedron is the amount of space it occupies. It is calculated by using the formula: Volume = (√2 / 12) × side³
Where ‘side’ is the length of any edge of the regular tetrahedron.
Volume of Tetrahedron Formula A regular tetrahedron has all edges of equal length.
To calculate its volume, you use the side length in the formula involving a constant factor.
The formula for the volume of a regular tetrahedron is given as follows: Volume = (√2 / 12) × side³
How to Derive the Volume of a Tetrahedron?
To derive the volume of a regular tetrahedron, we use the concept of volume as the total space occupied by a 3D object.
Given that all sides are equal, the volume can be derived using the height and base area of the pyramid shape: The general formula for the volume of a pyramid is: Volume = (1/3) × Base Area × Height
For a regular tetrahedron, the base is an equilateral triangle, and its height can be derived using geometry, leading to the formula: Volume = (1/3) × Base Area × Height
How to find the volume of a tetrahedron?
The volume of a tetrahedron is always expressed in cubic units, for example, cubic centimeters (cm³), cubic meters (m³).
Use the side length and the formula to find the volume. Let’s look at the formula for finding the volume of a tetrahedron: Write down the formula Volume = (√2 / 12) × side³
The side is the length of one edge of the tetrahedron. The side length of a tetrahedron is the length of one of its edges.
This is the only measurement needed to calculate the volume because all the sides of a regular tetrahedron are equal.
Once we know the length of the side, substitute that value for ‘side’ in the formula. To find the volume, apply the formula: Volume = (√2 / 12) × side³
Tips and Tricks for Calculating the Volume of Tetrahedron
Remember the formula: The formula for the volume of a tetrahedron is:Volume = (√2 / 12) × side³
Break it down: The volume is how much space fits inside the tetrahedron.
Since all the sides are equal, you need to use the formula involving the side length and constant factor.
Simplify the numbers: If the side length is a simple number, it is easier to calculate.
Check for cube roots If you are given the volume and need to find the side length, you can find the cube root and adjust for the constant factor.
Common Mistakes and How to Avoid Them in Volume of Tetrahedron
Making mistakes while learning the volume of the tetrahedron is common. Let’s look at some common mistakes and how to avoid them to get a better understanding of the volume of tetrahedrons.
Mistake 1
Confusing Volume with Surface Area
Some students confuse the formula for volume with the formula for surface area.
Surface area is calculated differently, involving the areas of the triangular faces, but volume uses the cube of the side length with a specific constant.
For example, the volume is Volume = (√2 / 12) × side³
, not the surface area formula.
Mistake 2
Confusing Volume with Perimeter
Some kids may think of the tetrahedron’s perimeter instead of the volume formula.
Volume is the space inside the tetrahedron, 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 pyramids
Some kids use the formula for the volume of a different type of pyramid instead of the tetrahedron formula.
Mistake 4
Confusing cubic volume with linear volume
Thinking of volume in terms of linear measurements. This happens when someone uses the side length (a linear measurement) instead of understanding that volume relates to cubic measurements.
Mistake 5
Incorrectly calculating the side length
Some students calculate the given volume without solving for the side length.
For example, if the volume is given and they need to find the side, they might forget to adjust for the constant factor after taking the cube root.
Volume of Tetrahedron Examples
Problem 1
A regular tetrahedron has a side length of 3 cm. What is its volume?
The volume of the tetrahedron is approximately 3.18 cm³.
Explanation
To find the volume of a tetrahedron, use the formula: V = (√2 / 12) × side³
Here, the side length is 3 cm, so:
V = (√2 / 12) × 3³
= (√2 / 12) × 27
≈ 3.18 cm³
Problem 2
A regular tetrahedron has a side length of 5 m. Find its volume.
The volume of the tetrahedron is approximately 14.73 m³.
Explanation
To find the volume of a tetrahedron, use the formula: V = (√2 / 12) × side³
Substitute the side length (5 m):
V = (√2 / 12) × 5³
= (√2 / 12) × 125
≈ 14.73 m³
Problem 3
The volume of a regular tetrahedron is 10 cm³. What is the side length of the tetrahedron?
The side length of the tetrahedron is approximately 4.14 cm.
Explanation
If you know the volume of the tetrahedron and need to find the side length, you’ll take the cube root of the adjusted volume.
The formula for the side length \( s \) is: s = ((12 × Volume) / √2)^(1/3)
s ≈ 4.14 cm
Problem 4
A regular tetrahedron has a side length of 2.5 inches. Find its volume.
The volume of the tetrahedron is approximately 1.48 inches³.
Explanation
Using the formula for volume:
V = (√2 / 12) × side³
Substitute the side length 2.5 inches:
V = (√2 / 12) × 2.5^3
V ≈ 1.48 inches³
Problem 5
You have a regular tetrahedron with a side length of 4 feet. How much space (in cubic feet) is available inside the tetrahedron?
The tetrahedron has a volume of approximately 7.54 cubic feet.
Explanation
Using the formula for volume:
V = (√2 / 12) × side³]
Substitute the side length 4 feet:
V = (√2 / 12) × 4^3 ≈ 7.54 ft³
FAQs on Volume of Tetrahedron
1.Is the volume of a tetrahedron the same as the surface area?
2.How do you find the volume if the side length is given?
3.What if I have the volume and need to find the side length?
4.Can the side length be a decimal or fraction?
5.Is the volume of a tetrahedron the same as the surface area?
Important Glossaries for Volume of Tetrahedron
- Side: The length of one of the tetrahedron’s edges. Since all edges of a regular tetrahedron are equal, the side length is the same for each edge.
- Volume: The amount of space enclosed within a 3D object. In the case of a tetrahedron, the volume is calculated using the side length and a constant factor. It is expressed in cubic units (e.g., cm³, m³).
- Regular Tetrahedron: A polyhedron with four triangular faces, all of which are equilateral triangles.
- Cubic Units: The units of measurement used for volume. If the side length is in centimeters (cm), the volume will be in cubic centimeters (cm³); if in meters, it will be in cubic meters (m³).
- Formula: A mathematical expression that calculates the volume of the tetrahedron, given by V = (√2 / 12) × side3
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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
