Parallelepiped Volume Calculator

A parallelepiped is a 3D shape with six parallelogram faces in three parallel pairs. Think of a box that has been sheared, so the sides no longer stand square to the base.

This calculator assumes a right parallelepiped, which is the common case in practice: the base is a parallelogram with sides a and b meeting at angle θ, and the third edge h stands 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: a leaning display block A display pedestal with a parallelogram base: a = 60 cm, b = 80 cm, included angle θ = 70° so it leans slightly, and h = 120 cm tall. Base area: 60 × 80 × sin(70°) = 60 × 80 × 0.9397 ≈ 4,510.5 cm². Volume: 4,510.5 × 120 ≈ 541,260 cm³ ≈ 0.54 m³.

Squared up to 90°, the same edges would give 60 × 80 × 120 = 576,000 cm³, so the 20° lean costs about 6% of the interior for no change at all in the amount of edge material.

In solid oak at 720 kg/m³ the block would mass around 390 kg, which is far too heavy to move. Real ones are hollow with reinforced corners, and then the number you want is the surface area, not this.

Where parallelepiped volumes matter:

• Crystal mineralogy. Calcite is the classic case: it cleaves into rhombohedra, parallelepipeds whose faces meet at about 75° and 105° rather than square. Gypsum and many other monoclinic minerals do the same. Halite is the counterexample, cleaving into true cubes, which are parallelepipeds too.
• 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.
• A stack of paper that has slid sideways. The leaning pile is an oblique parallelepiped, and it is the everyday demonstration of Cavalieri’s principle: nudging the sheets sideways changes the shape but not the paper, so the volume is identical to the tidy stack it started as.

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)

Set two of those angles to 90°, meaning one edge stands perpendicular to the base, and the whole radical collapses to sin(θ). That is exactly the case this calculator handles. For the general shape you need all three angles, which is normal in crystallography and rare anywhere else.

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 simply the most general boxy shape there is, and everything else on that list is a constrained version of it.

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). ✓