Parallelepiped Calculator

A parallelepiped is a three-dimensional solid bounded by six parallelograms, where opposite faces are parallel and congruent. It is the 3D analog of a parallelogram and generalizes shapes like cuboids and rhombohedra. It has 8 vertices, 12 edges, and 3 pairs of parallel faces.

The volume of a parallelepiped is calculated using V = a·b·c·√(1 − cos²α − cos²β − cos²γ + 2·cosα·cosβ·cosγ), where a, b, c are the side lengths and α, β, γ are the angles between the sides. This formula derives from the scalar triple product of the three edge vectors.

If you know the three edge lengths and the three included angles, substitute them into the formula V = a·b·c·√(1 − cos²α − cos²β − cos²γ + 2·cosα·cosβ·cosγ). If the angles are all 90°, the shape is a rectangular box and the volume simplifies to V = a·b·c.

The total surface area is the sum of the areas of all six faces. Since opposite faces are congruent, you compute the area of each pair: face ab = a·b·sin(γ), face ac = a·c·sin(β), and face bc = b·c·sin(α). The total surface area is 2·(face_ab + face_ac + face_bc).

When all three angles are 90°, the parallelepiped becomes a rectangular box (cuboid). When all sides are equal and all angles are equal, it becomes a rhombohedron. When all sides and angles are equal to 90°, it becomes a cube.

A parallelepiped has three pairs of parallel faces, giving it six faces in total. Each pair of opposite faces is parallel and congruent in shape and area.

Yes — if the three vectors defining the parallelepiped are coplanar (all lying in the same plane), the volume collapses to zero. This happens when the expression under the square root in the volume formula equals zero, meaning the vectors are linearly dependent.

A prism has two identical parallel polygonal bases connected by rectangular lateral faces, while a parallelepiped has six parallelogram faces with all opposite faces parallel. A rectangular parallelepiped (all 90° angles) is also a right prism with a parallelogram base.