Math
Volume of a Parallelepiped Calculator
Enter three edge vectors that meet at a corner, or the four vertices that define them, and mathcheck evaluates the scalar triple product a · (b × c) as a 3×3 determinant with every cofactor shown.
The absolute value is the parallelepiped's volume; one sixth of it is the volume of the tetrahedron with the same three edges. You also get the base area |b × c|, the height, the orientation (right- or left-handed), and a warning when the vectors are coplanar.
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Result
Parallelepiped volume
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- Scalar triple product a · (b × c)
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- Tetrahedron volume
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- b × c
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- Base area |b × c|
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- Height
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- Orientation
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Volume from the scalar triple product
A parallelepiped is a slanted box whose six faces are parallelograms. If a, b, and c are the three edges meeting at one corner, then b × c is a A quantity with both a size and a direction, often drawn as an arrow. perpendicular to the The number being raised to a power, or the side a shape's height is measured from. with length equal to the base area, and dotting it with a multiplies that area by the box's height. The result, a · (b × c), is the scalar triple product; its How far a number sits from zero, so the answer is never negative. is the volume, and it equals the One number computed from a square matrix that tells you whether it can be inverted. of the 3×3 A rectangular grid of numbers arranged in rows and columns. with a, b, c as rows.
a = ⟨1, 2, 3⟩, b = ⟨4, 5, 6⟩, c = ⟨7, 8, 10⟩
b × c = ⟨5·10 - 6·8, 6·7 - 4·10, 4·8 - 5·7⟩ = ⟨2, 2, -3⟩. a · (b × c) = 2 + 4 - 9 = -3. Volume 3, tetrahedron volume 1/2. The negative sign means (a, b, c) is a left-handed triple.
| a · (b × c) | Meaning |
|---|---|
| positive | a, b, c form a right-handed set |
| negative | left-handed set (swap any two vectors to flip the sign) |
| zero | the vectors are coplanar; no volume |
How to use this calculator
- Choose whether you have three edge vectors or four The turning point of a parabola, or a corner point of a shape..
- Enter the components. In vertex mode, P is the shared corner and the edges run from P to Q, R, and S.
- Read the volume in the header, and the signed triple product, tetrahedron volume, base area, height, and orientation below.
- Open Show the work to see b × c component by component, the Multiply two vectors piece by piece and add the results to get one number., and the A smaller determinant with a plus or minus sign attached, used to expand a bigger matrix. expansion of the determinant.
How to read the answer
The volume is the absolute value of the triple product; the sign only tells you the handedness of the ordered triple. A value of 0 means the three vectors lie in a common plane, which is also the test for linear dependence of three vectors in space. The tetrahedron volume is exactly one sixth of the parallelepiped, and the height is the volume divided by the base area |b × c|.
Common mistakes and edge cases
- Reporting a negative volume. Take the absolute value of the triple product.
- Forgetting the minus sign on the middle cofactor of the determinant.
- Using vertices instead of edge vectors. Subtract the shared corner first.
- Dividing by 3 for a tetrahedron; the factor is 6 (1/2 for the triangular base times 1/3 for the pyramid).
- Concluding that a zero triple product is an error. It means the vectors are coplanar, which is often the point of the question.
Worked examples
Textbook vectors
⟨1, 2, 3⟩, ⟨4, 5, 6⟩, ⟨7, 8, 10⟩
Parallelepiped volume
3
Unit cube
The standard basis vectors
Parallelepiped volume
1
Coplanar vectors
⟨1, 2, 3⟩, ⟨4, 5, 6⟩, ⟨7, 8, 9⟩
Parallelepiped volume
0
Left-handed order
Swapping b and c negates the triple product
Parallelepiped volume
1
Decimal components
⟨0.5, 0, 0⟩, ⟨0, 2.5, 0⟩, ⟨0.5, 0.5, 4⟩
Parallelepiped volume
5
From four vertices
P(0, 0, 0), Q(2, 0, 0), R(0, 3, 0), S(1, 1, 4)
Parallelepiped volume
24
Zero vector
Parallelepiped volume
0
Frequently asked questions
How do I find the volume of a parallelepiped from vectors?+
Compute the scalar triple product a · (b × c), or equivalently the determinant with a, b, c as rows, and take its absolute value.
Why is the tetrahedron volume one sixth?+
A tetrahedron on edges a, b, c has a triangular base of area |b × c|/2 and the pyramid formula adds another factor of 1/3, giving |a · (b × c)|/6.
What does a negative triple product mean?+
The ordered triple (a, b, c) is left-handed. The box is the same size; swapping any two of the vectors makes the product positive.
What if the triple product is zero?+
The vectors are coplanar: one of them lies in the plane of the other two. The parallelepiped collapses to a flat parallelogram and the vectors are linearly dependent.
Can I use four points instead of vectors?+
Yes. Pick one vertex P as the corner and use Q - P, R - P, and S - P as the edge vectors. The calculator's vertex mode does this subtraction for you.
Is a · (b × c) the same as (a × b) · c?+
Yes. The dot and cross can be swapped in the triple product, and cyclic reorderings a → b → c → a leave the value unchanged.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026