The parallelepiped volume calculator above finds the volume of a solid whose six faces are parallelograms — a box that has been sheared so its corners are no longer right angles. You can describe it either by the three edge vectors that meet at a corner, or by the three edge lengths and the three angles between them.
Arb Digital publishes it because the general case is genuinely different from the right-angled one. Our rectangular prism calculator covers boxes whose edges meet at right angles, where the volume is simply length times width times height. That formula is a special case of this page's, and it fails as soon as any angle departs from 90 degrees — set all three angles here to 90 and the two tools agree exactly, which is the boundary between them.
What This Parallelepiped Volume Calculator Does
It computes the volume via the scalar triple product, and reports that product with its sign preserved so you can read the orientation of the frame. It also gives the total surface area, computed as twice the sum of the three distinct face areas, and the length of the longest of the four space diagonals.
The breakdown lists the three face areas separately, which is useful because a sheared box has three different face shapes and the areas are rarely obvious from the edge lengths alone.
In angle mode the tool constructs a concrete set of vectors consistent with your lengths and angles and then runs the same arithmetic, so both modes go through identical code and the two cannot drift apart.
How to Use It
- Pick the input mode that matches the data you have. Neither is more accurate; they describe the same solid.
- In vector mode, enter the three edges as coordinates measured from the shared corner. Any consistent unit works, and the volume comes out in that unit cubed.
- In angle mode, be careful which angle is which: α sits between b and c, β between a and c, and γ between a and b.
- Check the orientation if you are working in graphics or crystallography, where handedness carries meaning.
- Read the volume as zero if the three edges turn out to be coplanar — the tool says so explicitly rather than reporting a rounding artefact.
The Formula and How It Is Calculated
With three edge vectors a, b and c sharing a corner, the volume is the absolute value of the scalar triple product: V = |a · (b × c)|. Wolfram MathWorld's parallelepiped page gives it in exactly that form, and its scalar triple product page notes that the product equals the determinant of the three vectors written as rows.
The geometric reading is worth carrying. The cross product b × c is a vector perpendicular to the base parallelogram whose magnitude equals that parallelogram's area. Dotting a into it projects a onto that perpendicular, which is the perpendicular height of the solid above the base. Area times height is volume, exactly as for a right box — the triple product is doing the projection that the right-angled case gets for free.
From lengths and angles the equivalent formula is V = abc × the square root of (1 − cos²α − cos²β − cos²γ + 2 cosα cosβ cosγ). This is the standard unit-cell volume expression used in crystallography, and it reduces to abc when all three cosines are zero. The square root argument can go negative for angle triples that no real solid can have, which the tool detects.
Not Every Set Of Angles Describes A Real Solid
This is the trap in angle mode and it catches people regularly. Three edge lengths can be anything positive, but three angles cannot be chosen freely — they must be geometrically consistent with each other, and most triples are not.
Try α = 20°, β = 20° and γ = 150°. Two of the edges are nearly aligned with the third but wildly separated from each other, which is impossible in three dimensions. The expression under the square root goes negative and there is no solid. The tool reports this in words rather than producing a NaN.
The formal condition is that the matrix of cosines between the three directions must be positive semi-definite, which is exactly the condition for those cosines to come from real vectors. In practice the quick check is the one the formula performs: compute the expression under the root and see whether it is positive. Zero means the three edges are coplanar and the solid is flat.
Where The Sign Of The Triple Product Matters
The volume is an absolute value, so the sign gets discarded. It is not meaningless, though, and two fields treat it as information.
In computer graphics the sign of a triple product determines whether a triangle faces towards or away from the camera, which is what back-face culling tests. It also determines whether a coordinate frame is right-handed or left-handed, a distinction that separates graphics APIs and file formats from each other and is a routine source of mirrored geometry.
In crystallography the convention is to describe unit cells with a right-handed axis set, and a negative triple product means the axes were listed in an order that reverses the handedness. The cell itself is fine; the labelling is not. Our cross product calculator and dot product calculator compute the two steps separately, and our determinant calculator evaluates the same quantity written as a 3×3 matrix.
The Volume Is Also A Scale Factor
There is a second reading of this number that is more general than geometry. Treat the three edge vectors as the columns of a 3×3 matrix, and that matrix is the linear map sending the unit cube to your parallelepiped. Its determinant is the triple product, so the volume is the factor by which that map multiplies every volume in space.
That is why a determinant of zero means a singular matrix and why the geometric picture is a flattened solid: the map has collapsed three dimensions into two or fewer, and no inverse can exist because the collapse destroyed information. It is also the Jacobian factor in a change of variables for a triple integral, which is the same quantity appearing in calculus rather than geometry.
The tetrahedron formed by the same three edges has exactly one sixth of this volume, which our tetrahedron volume calculator computes directly. That factor of six comes from the three-dimensional analogue of the fact that a triangle is half its enclosing parallelogram.
Where Slanted Boxes Actually Turn Up
Crystallography is the largest user. Six of the seven crystal systems have at least one non-right angle, and triclinic cells have none at all, so the general formula is the working one rather than a curiosity. Cell volume feeds directly into density calculations and into structure refinement.
Simulation software uses the same shape for periodic boundary conditions, where a non-orthogonal cell packs an anisotropic system more efficiently than a cubic one would. Geology uses it for the volume of material bounded by three non-perpendicular fault or bedding planes. And in graphics, an oriented bounding box under a shear transformation is a parallelepiped, and its volume is what tells you whether a transformation preserved scale. Converting the result between unit systems is the job of our volume converter, and our vector calculator handles the individual operations if you want to check a step.
Arb Digital builds calculators and interactive tools that state their assumptions and fail with a message rather than a blank field.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Multiplying the three edge lengths together — that only works when all three angles are right angles, and it overstates the volume otherwise.
- Pairing the angles with the wrong edges — α is between b and c, not between a and b, and swapping them changes the answer.
- Assuming any three angles are possible — most triples describe no real solid, and the formula reveals this by producing a negative value under the square root.
- Treating a negative triple product as a bug — it records the handedness of the edge order, and only its absolute value is the volume.
- Using half the surface area of a right box — a sheared box has three distinct face areas, and none of them is a simple product of two edge lengths.
Related Free Tools From Arb Digital
Handle the right-angled case with the rectangular prism calculator, take one sixth of the same volume with the tetrahedron volume calculator, evaluate the underlying determinant with the determinant calculator, compute the perpendicular with the cross product calculator, or change units with the volume converter. The free online tools hub lists every geometry tool we publish.
Frequently Asked Questions
The absolute value of the scalar triple product of the three edge vectors that meet at a corner: the magnitude of a dotted into b crossed with c, which equals the determinant of the three vectors.
A rectangular prism is the special case where all three edges meet at right angles. Then the triple product reduces to length times width times height. With any other angle that shortcut overstates the volume.
Yes. The volume is abc times the square root of one minus the three squared cosines plus twice their product, which is the standard crystallographic unit-cell formula.
Because most triples of angles are not geometrically consistent. The expression under the square root goes negative, which means no real solid has those three angles between its edges.
That the three edges, taken in the order given, form a left-handed frame rather than a right-handed one. The volume is the absolute value, so the sign records orientation only.
The three edge vectors are coplanar, so the solid is flat and has no volume. As a matrix the three vectors are linearly dependent and the determinant is zero.
A tetrahedron built on the same three edge vectors has exactly one sixth of the parallelepiped's volume, which is the three-dimensional analogue of a triangle being half its parallelogram.
This page explains a standard geometric formula for educational purposes. Results are dimensionless in the sense that the volume is expressed in whatever unit your edge lengths use, cubed.
