How this instrument works
A parallelepiped is a six-faced solid built by sweeping a parallelogram along a third direction — a generalized box whose three edge vectors meeting at one corner don't need to be mutually perpendicular the way a plain rectangular box's edges are. Its volume is found with the scalar triple product, V = |a · (b × c)|: the cross product b × c first finds a vector perpendicular to the base parallelogram formed by b and c, with a length equal to that parallelogram's own area, and the dot product with a then measures how much of a stands perpendicular to that base — exactly the 'height' a plain box-volume formula would multiply by.
When the three edge vectors happen to be mutually perpendicular, such as (2,0,0), (0,3,0), and (0,0,4), this formula reduces exactly to the familiar length × width × height, since the cross and dot products simplify away all the cross-terms in that special case. The scalar triple product is the general version of that same box-volume idea, valid for any three vectors, however skewed.
A result of exactly zero has a specific meaning: it means the three vectors are COPLANAR — they all lie flat in the same plane, with no genuine third dimension between them, so the 'solid' they would form has collapsed to a flat sheet with zero volume. This happens whenever one vector is a combination of the other two, such as one vector being an exact multiple of another.
- Enter the first edge vector's x, y, and z components into the Vector a fields.
- Enter the second edge vector's components into the Vector b fields.
- Enter the third edge vector's components into the Vector c fields.
- Read Volume: the sheet computes the cross product b × c, dots the result with a, and takes the absolute value.
Worked example — a 2×3×4 box built from perpendicular vectors
Three mutually perpendicular edge vectors, (2,0,0), (0,3,0), and (0,0,4), describe an ordinary rectangular box. The scalar triple product gives V = |2 × (3×4)| = 24, matching plain length × width × height exactly — a sanity check confirming the general formula reduces correctly to the familiar box formula whenever the edges happen to be perpendicular.
Now try edge vectors (1,2,0) and (2,4,0) — the second is exactly twice the first, so both lie along the same direction, unable to span a genuine parallelogram base no matter what the third vector is. The result is a volume of exactly 0, however the third vector is chosen — the flat, collapsed case the absolute-value formula reports honestly rather than hiding.
Questions
What is the formula for the volume of a parallelepiped?
V = |a · (b × c)|, the absolute value of the scalar triple product of its three edge vectors. The cross product b × c finds the base parallelogram's area as a perpendicular vector, and dotting with a measures how much of a stands perpendicular to that base.
How is a parallelepiped different from a rectangular box?
A rectangular box is the special case where all three edge vectors are mutually perpendicular. A general parallelepiped allows any three edge vectors, however slanted relative to each other, and the scalar triple product formula handles both cases — reducing exactly to length × width × height when the edges do happen to be perpendicular.
What does a volume of zero mean here?
It means the three edge vectors are coplanar — they all lie flat within a single plane, with no true third dimension separating them, so the shape they describe has collapsed to a flat sheet with no enclosed volume at all. This happens whenever one vector can be written as a combination of the other two.
Why is there an absolute value in the formula?
The raw scalar triple product a · (b × c) can come out negative depending on the order and orientation of the three vectors, but volume itself is always a non-negative quantity, so the absolute value strips away that sign, leaving only the physical size of the enclosed solid.
Does the order of a, b, and c matter?
The MAGNITUDE of the result doesn't change under a cyclic reordering of the three vectors (a, b, c gives the same volume as b, c, a), though the raw signed value before the absolute value can flip sign depending on orientation — which is exactly why the absolute value is taken at the end.