100 LearnersLast updated on July 20th, 2025
Volume of Parallelepiped
The volume of a parallelepiped is the total space it occupies or the number of cubic units it can hold. A parallelepiped is a 3D shape with six parallelogram faces. To find the volume of a parallelepiped, we use the scalar triple product of its vectors. In real life, kids can relate to the volume of a parallelepiped by thinking of things like a book, a box, or a shipping container. In this topic, let’s learn about the volume of a parallelepiped.
What is the volume of a parallelepiped?
The volume of a parallelepiped is the amount of space it occupies. It is calculated by using the formula:
Volume = |a·(b×c)| Where ‘a’, ‘b’, and ‘c’ are vectors representing the edges of the parallelepiped.
Volume of Parallelepiped Formula A parallelepiped is a 3-dimensional shape defined by vectors.
To calculate its volume, you take the dot product of one vector with the cross product of the other two.
The formula for the volume of a parallelepiped is given as follows: Volume = |a·(b×c)|
How to Derive the Volume of a Parallelepiped?
To derive the volume of a parallelepiped, we use the concept of the scalar triple product.
The formula for the volume of any parallelepiped is: Volume = |a·(b×c)| Here, 'a', 'b', and 'c' are vectors representing the edges of the parallelepiped.
The cross product (b×c) gives a vector perpendicular to the base, and the dot product with 'a' gives the height, leading to the volume.
How to find the volume of a parallelepiped?
The volume of a parallelepiped is always expressed in cubic units, for example, cubic centimeters (cm³), cubic meters (m³).
Use the scalar triple product to find the volume.
Let’s take a look at the formula for finding the volume of a parallelepiped: Write down the formula Volume = |a·(b×c)| Where 'a', 'b', and 'c' are vectors defining the edges.
Once you have the vectors, compute the cross product (b×c), then the dot product with 'a', and take the absolute value to find the volume.
Tips and Tricks for Calculating the Volume of Parallelepiped
Remember the formula: The formula for the volume of a parallelepiped is: Volume = |a·(b×c)|
Break it down: The volume is determined by the scalar triple product of vectors defining the edges.
Simplify the numbers: Calculate cross product first, then the dot product, and finally take the absolute value.
Check for orthogonality: If vectors 'b' and 'c' are orthogonal, the cross product simplifies, making calculations easier.
Common Mistakes and How to Avoid Them in Volume of Parallelepiped
Making mistakes while learning the volume of a parallelepiped is common.
Let’s look at some common mistakes and how to avoid them to get a better understanding of the volume of parallelepipeds.
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 faces of the parallelepiped, while volume is calculated by the scalar triple product.
For example, the volume is |a·(b×c)|, not the sum of face areas.
Mistake 2
Confusing Volume with Perimeter
Some students may think of the parallelepiped's perimeter instead of the volume formula.
Volume is the space inside the shape, whereas perimeter refers to the total length around the edges of a 2D shape in each face. Do not mix them up.
Mistake 3
Using the wrong formula for rectangular prisms
Some students use the formula for the volume of a rectangular prism (length × width × height) instead of the parallelepiped formula.
Mistake 4
Confusing scalar and vector products
Thinking of vector operations in terms of linear measurements.
This happens when someone uses just vector magnitudes instead of understanding that volume relates to the scalar triple product.
Mistake 5
Incorrectly calculating the cross product
Some students calculate the given volume without properly computing the cross product. Ensure you compute (b×c) first before proceeding to the dot product with 'a'.
Volume of Parallelepiped Examples
Problem 1
A parallelepiped is defined by vectors a = [1, 2, 3], b = [4, 0, 0], and c = [0, 5, 0]. What is its volume?
The volume of the parallelepiped is 20 cubic units.
Explanation
To find the volume of a parallelepiped, use the formula: V = |a·(b×c)| Compute b×c: [4, 0, 0]×[0, 5, 0] = [0, 0, 20] Then, a·(b×c) = [1, 2, 3]·[0, 0, 20] = 60 Volume = |60| = 60
Problem 2
A parallelepiped is defined by vectors a = [2, 0, 0], b = [0, 3, 0], and c = [0, 0, 4]. Find its volume.
The volume of the parallelepiped is 24 cubic units.
Explanation
To find the volume of a parallelepiped, use the formula: V = |a·(b×c)| Compute b×c: [0, 3, 0]×[0, 0, 4] = [12, 0, 0] Then, a·(b×c) = [2, 0, 0]·[12, 0, 0] = 24 Volume = |24| = 24
Problem 3
The volume of a parallelepiped defined by vectors a = [1, 1, 1], b = [0, 1, 0], and c = [1, 0, 1] is 2 cubic units. Is this correct?
Yes, the volume is 2 cubic units.
Explanation
Compute b×c: [0, 1, 0]×[1, 0, 1] = [1, 0, -1] Then, a·(b×c) = [1, 1, 1]·[1, 0, -1] = 0 Volume = |0| = 0 (Typo in question; the correct volume is 0).
Problem 4
A parallelepiped is defined by vectors a = [3, 3, 3], b = [1, 0, 0], and c = [0, 2, 0]. What is its volume?
The volume of the parallelepiped is 0 cubic units.
Explanation
Compute b×c: [1, 0, 0]×[0, 2, 0] = [0, 0, 2] Then, a·(b×c) = [3, 3, 3]·[0, 0, 2] = 6 Volume = |6| = 6
Problem 5
You have a parallelepiped defined by vectors a = [2, 0, 1], b = [1, 1, 1], and c = [0, 1, 1]. What is its volume?
The volume of the parallelepiped is 1 cubic unit.
Explanation
Compute b×c: [1, 1, 1]×[0, 1, 1] = [0, -1, 1] Then, a·(b×c) = [2, 0, 1]·[0, -1, 1] = 1 Volume = |1| = 1
FAQs on Volume of Parallelepiped
1.Is the volume of a parallelepiped the same as the surface area?
2.How do you find the volume if the vectors are given?
3.What if I have the volume and need to verify the vectors?
4.Can the vectors be in any orientation?
5.Can a parallelepiped have zero volume?
Important Glossaries for Volume of Parallelepiped
- Vector: A quantity with both magnitude and direction, used to define edges of a parallelepiped.
- Cross Product: A vector perpendicular to two given vectors, used in calculating volume.
- Dot Product: A scalar product of two vectors, part of the volume calculation.
- Scalar Triple Product: The product used to calculate the volume of a parallelepiped, given by |a·(b×c)|.
- Cubic Units: Units of measurement for volume, such as cubic centimeters (cm³) or cubic meters (m³).
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Seyed Ali Fathima S
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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.
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