Parallelepiped Volume Calculator
Compute the volume of a right parallelepiped from base sides, included angle, and height.
For oblique boxes, crystal forms, and lean-to structures.
A parallelepiped is a 3D shape with six parallelogram faces — three pairs of parallel faces. It’s like a box that’s been “sheared” so the sides aren’t perpendicular to the base.
This calculator assumes a right parallelepiped — the most common case in practice — where the base is a parallelogram (sides a and b at an angle θ to each other) and the third edge h is perpendicular to that base.
V = a × b × sin(θ) × h
The a × b × sin(θ) part is the parallelogram base area. Multiply by perpendicular height h to get volume.
Worked example — leaning architectural display block: A modern art display pedestal with a parallelogram base: a = 60 cm, b = 80 cm, included angle θ = 70° (so it leans slightly), h = 120 cm tall. Base area: 60 × 80 × sin(70°) ≈ 60 × 80 × 0.9397 ≈ 4,510.4 cm². Volume: 4,510.4 × 120 ≈ 541,250 cm³ ≈ 0.54 m³.
If made of solid oak (density 720 kg/m³): mass ≈ 390 kg. Way too heavy for a movable display — most are hollow with reinforced corners.
Where parallelepiped volumes matter:
- Crystal mineralogy. Many naturally occurring crystals (calcite, gypsum, halite) form parallelepiped habits with non-90° angles between edges.
- Architectural lean-to structures. Modern building wings that tilt outward or inward have parallelepiped floor plans.
- Oblique storage containers. Some industrial bins are designed at angles to nest with conveyor or chute systems.
- Crystallographic unit cells in chemistry. Crystal structures are described as parallelepipeds with lattice parameters a, b, c and angles α, β, γ.
- Stack of paper or magazines that has slid sideways. A “leaning tower” of magazines is a parallelepiped.
The “general” (oblique) parallelepiped — beyond this calculator:
A truly general parallelepiped has THREE edges at arbitrary angles to each other. Volume is given by the scalar triple product of the three edge vectors. If the three edges have lengths a, b, c and the angles between them are α, β, γ:
V = a × b × c × √(1 − cos²α − cos²β − cos²γ + 2 × cos α × cos β × cos γ)
This collapses to a × b × c × sin(θ) when one edge is perpendicular to the base — the case this calculator handles. For full generality, you’d need to know all three angles, which is unusual in practical problems.
Distinguishing the parallelepiped from related shapes:
- Rectangular prism (cuboid): all angles 90°. Special case where θ = 90° in the formula.
- Rhombohedron: all six faces are congruent rhombi. Special case.
- Cube: rhombohedron where all angles are 90° and edges are equal.
All three are parallelepipeds — the parallelepiped is the most general “boxy” shape.
Sanity check:
- θ = 90° (perpendicular base sides): V = a × b × h. Reduces to a rectangular prism. ✓
- θ = 0° or 180° (degenerate base): V = 0. ✓
- a = b = h and θ = 90°: V = a³ (cube). ✓