Parallelepiped Volume Calculator

What Is a Parallelepiped?

A parallelepiped is a three-dimensional shape with six parallelogram faces. When all angles are right angles, it becomes a rectangular prism. In vector geometry, three adjacent edge vectors that share a common starting point fully describe the shape.

Volume Formula Using Vectors

The volume equals the absolute value of the scalar triple product:

Here \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are three non-coplanar edge vectors. The cross product \(\vec{a} \times \vec{b}\) gives a vector perpendicular to both inputs, and the dot product with \(\vec{c}\) measures how far the shape extends along that direction.

Surface Area Formula

The total surface area combines the areas of all three unique parallelogram faces, counted twice:

Example With Three Vectors

Let \(\vec{a} = [1, 2, 3]\), \(\vec{b} = [5, -4, 7]\), and \(\vec{c} = [-5, 1, 12]\).

1. Compute \(\vec{a} \times \vec{b} = [34, -4, -26]\)
2. Compute \((\vec{a} \times \vec{b}) \cdot \vec{c} = -290\)
3. Volume: \(V = |-290| = 290\)

Related tools: Vector Calculator, Matrix Determinant Calculator, and Volume Calculator.