104 LearnersLast updated on July 11th, 2025
Volume of Octahedron
The volume of an octahedron is the total space it occupies or the number of cubic units it can hold. An octahedron is a 3D shape with eight equilateral triangular faces. To find the volume of a regular octahedron, we use the formula involving the side length of its triangles. In real life, an octahedron can be related to shapes like certain types of dice or crystals. In this topic, let’s learn about the volume of an octahedron.
What is the volume of the octahedron?
The volume of an octahedron is the amount of space it occupies. It is calculated by using the formula: Volume = (√2/3) × side³ Where 'side' is the length of any edge of the octahedron.
Volume of Octahedron Formula An octahedron is a 3-dimensional shape where all edges are equal in length.
To calculate its volume, we use the formula that involves the side length of the equilateral triangles forming the octahedron.
The formula for the volume of a regular octahedron is given as follows: Volume = (√2/3) × side³
How to Derive the Volume of an Octahedron?
To derive the volume of an octahedron, we use geometric principles related to its structure. An octahedron can be thought of as two pyramids with a common base.
The formula for the volume of a pyramid is: Volume = (1/3) × Base Area × Height
For a regular octahedron, the base area and height are related through geometric relationships, leading to: Volume = (√2/3) × side³
How to find the volume of an octahedron?
The volume of an octahedron is always expressed in cubic units, for example, cubic centimeters (cm³), cubic meters (m³).
Use the formula for finding the volume of an octahedron: Write down the formula Volume = (√2/3) × side³
The side is the length of one edge of the octahedron. Once we know the length of the side, substitute that value for 'side' in the formula Volume = (√2/3) × side³
To find the volume, use the calculation for side³, then multiply by √2/3.
Tips and Tricks for Calculating the Volume of Octahedron
Remember the formula: The formula for the volume of an octahedron is: Volume = (√2/3) × side³
Break it down: The volume is how much space fits inside the octahedron. Since all sides are equal, you just need to calculate the side length cubed and then multiply by √2/3.
Simplify the numbers: If the side length is a simple number like 2, 3, or 4, it is easier to perform the cube and multiplication calculations.
Check for cube roots: If you are given the volume and need to find the side length, you can find the cube root followed by a division by √2/3.
Common Mistakes and How to Avoid Them in Volume of Octahedron
Making mistakes while learning the volume of the octahedron is common. Let’s look at some common mistakes and how to avoid them to get a better understanding of the volume of octahedrons.
Mistake 1
Confusing Volume with Surface Area
Some students confuse the formula for volume with the formula for surface area. Surface area is calculated by 2 × √3 × side², but volume is calculated by using the side cubed and multiplying by √2/3.
Mistake 2
Confusing Volume with Perimeter
Some kids may think of the perimeter of the triangular faces instead of the volume formula. Volume is the space inside the octahedron, 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 polyhedra
Some kids use the formula for the volume of a cube or tetrahedron instead of the octahedron 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 (which is 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 with solving for the side length. For example, if the volume is given, they might forget to take the cube root of the volume and adjust by multiplying by 3/√2.
Volume of Octahedron Examples
Problem 1
An octahedron has a side length of 3 cm. What is its volume?
The volume of the octahedron is 6.928 cm³.
Explanation
To find the volume of an octahedron, use the formula: V = (√2/3) × side³
Here, the side length is 3 cm, so: V = (√2/3) × 3³ = (√2/3) × 27 = 6.928 cm³
Problem 2
An octahedron has a side length of 5 m. Find its volume.
The volume of the octahedron is 58.925 m³.
Explanation
To find the volume of an octahedron, use the formula: V = (√2/3) × side³
Substitute the side length (5 m): V = (√2/3) × 5³ = (√2/3) × 125 = 58.925 m³
Problem 3
The volume of an octahedron is 12 cm³. What is the side length of the octahedron?
The side length of the octahedron is approximately 2.289 cm.
Explanation
If you know the volume of the octahedron, and you need to find the side length, you’ll take the cube root of the volume divided by (√2/3). The formula for the side length s is: s = (volume × 3/√2)(1/3) s = (12 × 3/√2)(1/3) ≈ 2.289 cm
Problem 4
An octahedron has a side length of 2 inches. Find its volume.
The volume of the octahedron is 3.771 inches³.
Explanation
Using the formula for volume: V = (√2/3) × side³ Substitute the side length 2 inches: V = (√2/3) × 2³ = (√2/3) × 8 = 3.771 inches³
Problem 5
You have an octahedron-shaped crystal with a side length of 4 feet. How much space (in cubic feet) is available inside the crystal?
The crystal has a volume of 30.96 cubic feet.
Explanation
Using the formula for volume: V = (√2/3) × side³
Substitute the side length 4 feet: V = (√2/3) × 4³ = (√2/3) × 64 = 30.96 ft³
FAQs on Volume of Octahedron
1.Is the volume of an octahedron 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.How is the volume of an octahedron related to a tetrahedron?
Important Glossaries for Volume of Octahedron
- Side: The length of one of the octahedron’s edges. Since all edges of a regular octahedron are equal, the side length is the same for each edge.
- Volume: The amount of space enclosed within a 3D object. For an octahedron, the volume is calculated by the formula (√2/3) × side³.
- 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).
- Regular Octahedron: A three-dimensional shape with eight equilateral triangular faces, twelve edges of equal length, and six vertices.
- Equilateral Triangle: A triangle where all three sides are of equal length, commonly used in the faces of a regular octahedron.
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
