Free Volume of a Parallelepiped Calculator

Understanding the Parallelepiped Volume Formula

A parallelepiped is a six‑faced polyhedron where every face is a parallelogram. When all angles are right angles it becomes a rectangular prism, but in the general case you need three adjacent edges (or three vectors) and the angles between them to describe it completely.

This vector volume calculator – also functioning as a scalar triple product calculator – lets you obtain the volume and the parallelepiped surface area from three vectors, four vertices, or edge lengths with the corresponding angles.

Volume via Scalar Triple Product

The fundamental parallelepiped volume formula uses the absolute value of the scalar triple product:

$$V = \left| (\vec{a} \times \vec{b}) \cdot \vec{c} \right|$$

Here $\vec{a},\vec{b},\vec{c}$ are three vectors representing the three adjacent sides that meet at one vertex.
The cross product $\vec{a} \times \vec{b}$ yields a vector whose magnitude equals the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$, and the subsequent dot product with $\vec{c}$ projects that base area onto the direction of $\vec{c}$ – effectively multiplying base area by height.

The same volume can be expressed as the absolute value of a determinant:

$$V = \left| \det\begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{pmatrix} \right|$$

where $a_i, b_i, c_i$ are the components of the three vectors.

From Four Vertices

If only the coordinates of four vertices (e.g., $P,Q,R,S$) are known, construct three vectors from one vertex – for instance $\vec{a} = Q - P$, $\vec{b} = R - P$, $\vec{c} = S - P$ – then apply the scalar triple product.

Volume from Edge Lengths and Angles

When the edge lengths $a,b,c$ and the angles between them are given, the volume is computed with the formula:

$$V = abc\sqrt{1 + 2\cos\alpha\cos\beta\cos\gamma - \cos^{2}\alpha - \cos^{2}\beta - \cos^{2}\gamma}$$

where:

• $\alpha$ – angle between edges $b$ and $c$
• $\beta$ – angle between edges $a$ and $c$
• $\gamma$ – angle between edges $a$ and $b$

If all edges are equal and all angles are $90^\circ$, this reduces to the formula for a cube.

Surface Area of a Parallelepiped

You can compute the surface area in two equivalent ways:

Using vectors –

$$A = 2\left(|\vec{a} \times \vec{b}| + |\vec{b} \times \vec{c}| + |\vec{a} \times \vec{c}|\right)$$

Using edge lengths and angles –

$$A = 2\left(ab\sin\gamma + bc\sin\alpha + ac\sin\beta\right)$$

Both formulas add the areas of the six parallelogram faces (three pairs of congruent faces).

How to Use This 3D Vector Cross Product and Volume Tool

The calculator provides three input modes, selectable from a drop‑down menu:

1. Vectors mode – enter the components of $\vec{a},\vec{b},\vec{c}$. The volume and surface area are displayed immediately.
2. Vertices mode – input the coordinates of four points (P, Q, R, S). The calculator derives the necessary vectors and computes the results.
3. Edge lengths & angles mode – supply the lengths $a,b,c$ and the three angles $\alpha,\beta,\gamma$ (in degrees or radians). Volume and area are calculated using the angle‑based formulas.

If the input is rejected because conflicting values from another mode remain, use the Clear all changes or Reload calculator button to reset the tool.

What If the Scalar Triple Product Is Zero?

When $V = 0$, the three vectors are coplanar – they lie in the same plane. Similarly, if the surface area formula returns zero, the vectors are collinear. A zero volume indicates a degenerate or flat parallelepiped.

For a quick check: a tetrahedron built from the same three vectors has a volume exactly one‑sixth of the parallelepiped’s volume.