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105 LearnersLast updated on September 11, 2025
Volume of a Parallelepiped Calculator
Calculators are reliable tools for solving simple mathematical problems and advanced calculations like geometry. Whether you’re studying, engineering, or planning a construction project, calculators will make your life easy. In this topic, we are going to talk about the volume of a parallelepiped calculator.
What is Volume of a Parallelepiped Calculator?
A volume of a parallelepiped calculator is a tool to figure out the volume of a parallelepiped using its edge vectors.
Since a parallelepiped is a three-dimensional figure with parallelogram faces, the calculator helps compute its volume from the vectors.
This calculator makes the calculation much easier and faster, saving time and effort.
How to Use the Volume of a Parallelepiped Calculator?
Given below is a step-by-step process on how to use the calculator:
Step 1: Enter the vectors: Input the three vectors defining the edges of the parallelepiped into the given fields.
Step 2: Click on calculate: Click on the calculate button to get the volume.
Step 3: View the result: The calculator will display the result instantly.
How to Calculate the Volume of a Parallelepiped?
To calculate the volume of a parallelepiped, we use the scalar triple product of its edge vectors a, b, and c.
The formula is: Volume = |a · (b × c)| The scalar triple product involves taking the cross product of two vectors and then the dot product with the third vector.
This gives the absolute value of the volume of the parallelepiped.
Tips and Tricks for Using the Volume of a Parallelepiped Calculator
When using a volume of a parallelepiped calculator, there are a few tips and tricks that we can use to make it easier and avoid mistakes:
Visualize the vectors in 3D space to understand their orientation.
Remember that the order of vectors matters in cross and dot products.
Use decimal precision for more accurate results.
Common Mistakes and How to Avoid Them When Using the Volume of a Parallelepiped Calculator
We might think that when using a calculator, mistakes will not happen. But it's possible to make mistakes when using a calculator.
Mistake 1
Rounding too early before completing the calculation.
Wait until the very end for a more accurate result.
For example, you might round 5.29 to 5 before finishing the calculation, but this will be incorrect. You need to remember the decimal part.
Mistake 2
Forgetting the absolute value in the scalar triple product
After calculating the scalar triple product, remember to take the absolute value to ensure the volume is non-negative.
Mistake 3
Incorrectly computing the cross product or dot product
Ensure you correctly perform vector operations. Errors in cross or dot products can lead to incorrect volume results.
Mistake 4
Relying on the calculator a bit too much for precision
When using calculators, remember that the result is an estimate and may need adjustment for real-life applications.
Mistake 5
Assuming all calculators will handle all scenarios
Not all calculators account for vector properties or provide detailed steps. Double-check calculations if needed.
Volume of a Parallelepiped Calculator Examples
Problem 1
What is the volume of a parallelepiped with edge vectors a = (1, 2, 3), b = (4, 5, 6), and c = (7, 8, 9)?
Use the formula:
Volume = |a · (b × c)| b × c = (5*9 - 6*8, 6*7 - 4*9, 4*8 - 5*7) = (9, -6, -3) a · (b × c) = 1*9 + 2*(-6) + 3*(-3) = 9 - 12 - 9 = -12
Volume = |-12| = 12
Explanation
The cross product of b and c is calculated first. Then, the dot product with a gives -12, and the absolute value is taken for the volume.
Problem 2
How do you find the volume of a parallelepiped with vectors a = (2, 0, 1), b = (1, 1, 1), c = (0, 2, 3)?
Use the formula:
Volume = |a · (b × c)| b × c = (1*3 - 1*2, 1*0 - 3*1, 1*2 - 1*0) = (1, -3, 2) a · (b × c) = 2*1 + 0*(-3) + 1*2 = 2 + 0 + 2 = 4
Volume = |4| = 4
Explanation
The cross product of b and c is found, and then the dot product with a results in the volume of 4.
Problem 3
Find the volume of a parallelepiped with vectors a = (3, 4, 5), b = (5, 6, 7), c = (6, 7, 8).
Use the formula:
Volume = |a · (b × c)| b × c = (6*8 - 7*7, 7*6 - 5*8, 5*7 - 6*6) = (4, 2, -1) a · (b × c) = 3*4 + 4*2 + 5*(-1) = 12 + 8 - 5 = 15
Volume = |15| = 15
Explanation
The cross product of b and c is calculated, followed by the dot product with a, giving a volume of 15.
Problem 4
What is the volume of a parallelepiped with vectors a = (1, 0, 0), b = (0, 1, 0), c = (0, 0, 1)?
Use the formula:
Volume = |a · (b × c)| b × c = (0, 0, 1) a · (b × c) = 1*1 = 1
Volume = |1| = 1
Explanation
The vectors are orthogonal and form the basis of a unit cube, so the volume is 1.
Problem 5
Calculate the volume of a parallelepiped with vectors a = (2, 3, 4), b = (4, 5, 6), c = (0, 1, 1).
Use the formula:
Volume = |a · (b × c)| b × c = (5*1 - 6*1, 6*0 - 4*1, 4*1 - 5*0) = (-1, -4, 4) a · (b × c) = 2*(-1) + 3*(-4) + 4*4 = -2 - 12 + 16 = 2
Volume = |2| = 2
Explanation
The cross product of b and c is calculated, followed by the dot product with a, resulting in a volume of 2.
FAQs on Using the Volume of a Parallelepiped Calculator
1.How do you calculate the volume of a parallelepiped?
2.What is the scalar triple product?
3.Why is the volume given by the absolute value of the scalar triple product?
4.How do I use a volume of a parallelepiped calculator?
5.Is the volume of a parallelepiped calculator accurate?
Glossary of Terms for the Volume of a Parallelepiped Calculator
- Vectors: Mathematical entities with both magnitude and direction, represented in 3D space by coordinates.
- Cross Product: A binary operation on two vectors in three-dimensional space resulting in a vector perpendicular to both.
- Dot Product: An algebraic operation that takes two equal-length sequences of numbers and returns a single number.
- Scalar Triple Product: The result of a dot product of one vector with the cross product of two others, used to find volume.
- Parallelepiped: A six-faced figure (also called a polyhedron) where each face is a parallelogram.
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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
