Volume of a Parallelepiped Calculator
Find the volume and surface area of any parallelepiped — an oblique box with three distinct edge lengths (a, b, c) and three pairwise angles (α, β, γ) between them. Setting all angles to 90° gives a rectangular box.
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V = abc × √(1 − cos²α − cos²β − cos²γ + 2cosα cosβ cosγ)
V = abc√det(G)
60 units³- 1
Cosines of the three angles
cos α = 0, cos β = 0, cos γ = 0 - 2
Gram determinant
1 − 0² − 0² − 0² + 2 × 0 × 0 × 0 = 1Must be positive for a valid 3D shape to exist. - 3
√(Gram determinant)
√1 = 1 - 4
Volume
5 × 4 × 3 × 1 = 60
How does this calculator work?
Parallelepiped volume V = a·b·c × √(1 − cos²α − cos²β − cos²γ + 2cosα cosβ cosγ), where a, b, c are edge lengths and α, β, γ are the angles between (b,c), (a,c), (a,b). For a rectangular box all angles are 90° and V = abc. Face areas = edge₁ × edge₂ × sin(included angle); total surface area = 2(bc sinα + ac sinβ + ab sinγ).
Formula
How this is calculated
A parallelepiped is a three-dimensional figure with six parallelogram faces, defined by three edge vectors from one vertex. When those edges are mutually perpendicular the shape is a rectangular box and the volume simplifies to abc. When the edges meet at oblique angles the volume is reduced by a factor that depends on those angles.
The volume formula follows from the scalar triple product: V = |u · (v × w)|. Expanding for edge lengths a, b, c and pairwise angles α (between b and c), β (between a and c), γ (between a and b) gives V = abc × √(1 − cos²α − cos²β − cos²γ + 2cosα cosβ cosγ). The expression under the root is the Gram determinant — it equals 1 for a rectangular box and decreases toward 0 as the shape flattens. A non-positive Gram determinant means the angle combination is geometrically impossible.
Each of the six faces is a parallelogram with area equal to the product of its two edge lengths times the sine of the included angle: face(b,c) = bc sinα, face(a,c) = ac sinβ, face(a,b) = ab sinγ. There are two copies of each face, so total surface area = 2(bc sinα + ac sinβ + ab sinγ). All inputs share one consistent length unit — the volume is in cubic units and areas in square units.
Frequently asked questions
When all three angles are 90° — edges meet at right angles — cosines are zero and √det(G) = 1, giving V = abc, the standard cuboid formula. You can verify this by entering 90° for all three angles.
The Gram determinant (1 − cos²α − cos²β − cos²γ + 2cosα cosβ cosγ) must be positive for a real 3D shape to exist. Very acute or obtuse angle combinations can make it zero or negative. For example, all three angles equal 0° or 180° are degenerate (flat) cases.
A prism has two identical polygonal bases connected by rectangular lateral faces. A parallelepiped has six parallelogram faces — opposite pairs are equal and parallel. Every rectangular box is both, but an oblique parallelepiped (angles ≠ 90°) is not a standard prism.
Also known as
TG we-Calculate Editorial Team. (2026). Volume of a Parallelepiped Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/volume-of-a-parallelepiped-calculator
TG we-Calculate Editorial Team. "Volume of a Parallelepiped Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/volume-of-a-parallelepiped-calculator.
TG we-Calculate Editorial Team, "Volume of a Parallelepiped Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/volume-of-a-parallelepiped-calculator
@misc{wecalculate_volume_of_a_parallelepiped_calculator, title = {Volume of a Parallelepiped Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/volume-of-a-parallelepiped-calculator}}, year = {2026}, note = {TG we-Calculate} }
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