Calculate the volume of a parallelepiped using the scalar triple product of 3 edge vectors. Also find surface area, space diagonal, face diagonals, and edge angles.
A parallelepiped is a three-dimensional solid formed by six parallelogram faces — the 3D generalization of a parallelogram. It is defined by three edge vectors emanating from a single vertex. The volume of a parallelepiped is computed elegantly using the scalar triple product: V = |a⃗ · (b⃗ × c⃗)|, which is also the absolute value of the determinant of the 3×3 matrix formed by the three vectors.
This formula is fundamental in linear algebra and physics. It appears in change-of-variable formulas for multiple integrals (the Jacobian determinant), in crystallography (unit cell volumes), and in mechanics (torque and angular momentum). A right rectangular box (cuboid) is the special case where all three edge vectors are mutually perpendicular.
This calculator takes three edge vectors in 3D and computes: volume, surface area, space diagonal, all three face diagonals, face areas, and angles between edges. It shows the full step-by-step computation including the cross product and dot product. Presets for a unit cube, orthogonal box, and various oblique parallelepipeds let you explore how volume changes with edge angles.
Computing the scalar triple product by hand involves a cross product (6 multiplications and 3 subtractions) followed by a dot product (3 multiplications and 2 additions). Getting any sign wrong invalidates the result. Add surface area, diagonals, and angles, and you have a dozen separate computations.
This calculator performs everything instantly with full step-by-step visibility, making it ideal for linear algebra students, physics students computing volumes or flux, and engineers working with oblique coordinate systems.
Volume = |a⃗ · (b⃗ × c⃗)| (scalar triple product = absolute value of 3×3 determinant). Surface area = 2(|a⃗×b⃗| + |b⃗×c⃗| + |a⃗×c⃗|). Space diagonal = |a⃗ + b⃗ + c⃗|. Face diagonal = |u⃗ + v⃗| for adjacent edges u⃗, v⃗. Angle between edges = arccos(u⃗·v⃗/(|u⃗||v⃗|)).
Result: Volume = 24, Surface Area = 52, Space Diagonal = 5.39
For vectors a⃗=⟨2,0,0⟩, b⃗=⟨0,3,0⟩, c⃗=⟨0,0,4⟩: b⃗×c⃗ = ⟨12,0,0⟩. a⃗·(b⃗×c⃗) = 24. This is a right rectangular box 2×3×4. SA = 2(6+12+8) = 52. Space diag = √(4+9+16) = √29 ≈ 5.39.
The scalar triple product appears throughout physics. In electromagnetism, the magnetic flux through a parallelogram-shaped surface involves the cross product. In mechanics, the scalar triple product determines whether three force vectors can produce a net torque. In fluid dynamics, it measures the volume flow rate through a tilted surface element.
The cyclic property a⃗·(b⃗×c⃗) = b⃗·(c⃗×a⃗) = c⃗·(a⃗×b⃗) means you can cyclically permute the vectors without changing the result — a useful symmetry in many physical derivations.
The connection between determinants and volume is one of the most beautiful results in linear algebra. The absolute value of the n×n determinant of a matrix gives the n-dimensional volume (hypervolume) of the parallelepiped spanned by its column vectors. For n=2, this is the area of a parallelogram; for n=3, the volume of a parallelepiped; for higher dimensions, it generalizes seamlessly.
This perspective also explains why the determinant is zero when vectors are linearly dependent: a degenerate parallelepiped has zero volume in the full-dimensional space.
In crystallography, the unit cell of a crystal is a parallelepiped defined by three lattice vectors. The volume of this unit cell, computed via the scalar triple product, is fundamental to determining crystal density, X-ray diffraction patterns, and material properties. The Bravais lattices classify all possible unit cell geometries, from cubic (all edges equal, all angles 90°) to triclinic (all different, no right angles).
A parallelepiped is a 3D solid with six parallelogram faces. It is the 3D analogue of a parallelogram and is defined by three non-coplanar edge vectors from a single vertex.
The scalar triple product a⃗·(b⃗×c⃗) is a scalar value equal to the signed volume of the parallelepiped. It equals the determinant of the 3×3 matrix [a⃗ b⃗ c⃗].
The volume is zero when the three vectors are coplanar (linearly dependent). This means the "parallelepiped" has collapsed to a flat shape with no 3D extent.
The scalar triple product is exactly the determinant of the 3×3 matrix whose columns (or rows) are the three vectors. The absolute value of the determinant gives the volume.
A right parallelepiped has all edge pairs meeting at right angles (dot products = 0). This is equivalent to a rectangular box (cuboid).
In multivariable calculus, changing variables in a triple integral introduces a Jacobian determinant factor — which is exactly the scalar triple product of the transformed basis vectors. This is why the triple product "measures volume."