Free Volume of a Parallelepiped Calculator

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A parallelepiped is a six-faced 3D solid built from three edge vectors that all share a common starting vertex — think of it as a "squashed" box whose faces are parallelograms rather than necessarily rectangles. Its volume is found using the scalar triple product of the three edge vectors, a·(b×c): first taking the cross product of two edges (which gives a vector perpendicular to both, with a magnitude equal to the area of the parallelogram they form), then dotting the result with the third edge (which projects that third edge onto the perpendicular direction and scales by the base area). This construction connects directly to determinants in linear algebra (the scalar triple product is exactly the determinant of the 3×3 matrix formed by the three vectors' components), to crystallography (the unit cell of a crystal lattice is a parallelepiped, and its volume from the lattice vectors determines the material's density), and to physics and engineering problems involving three non-coplanar forces or displacements, where the enclosed volume indicates how "independent" the three directions truly are.

How it works

Enter the three edge vectors a, b, and c, each extending from the same shared vertex of the solid. The calculator first computes the cross product b×c, a new vector perpendicular to both b and c whose length equals the area of the parallelogram those two vectors span. It then takes the dot product of a with that result, a·(b×c), which measures how much of vector a points along that perpendicular direction, scaled by the base area — giving exactly the enclosed volume. Because this triple product can come out negative depending on the vectors' handedness (whether a, b, c form a "right-handed" or "left-handed" set), the calculator reports the absolute value, since a physical volume can never be negative.

Enter vector a — x.
Enter vector a — y.
Enter vector a — z.
Enter vector b — x.
Enter vector b — y.
Enter vector b — z.
Enter vector c — x.
Enter vector c — y.
Enter vector c — z.
Click Calculate to see your results.

Examples

a=(1,0,0), b=(0,1,0), c=(0,0,1)

These are the three unit axis vectors, forming an ordinary unit cube. b×c = (1,0,0), and a·(1,0,0) = 1, so the volume is 1 — matching the obvious 1×1×1 cube volume.

a=(2,0,0), b=(0,3,0), c=(0,0,4)

b×c = (3×4−0×0, 0×0−0×4, 0×0−3×0) = (12, 0, 0). Then a·(12,0,0) = 2×12 = 24, matching the rectangular box volume of 2×3×4 = 24 directly, since these vectors are mutually perpendicular.

a=(1,2,3), b=(0,1,4), c=(5,6,0)

b×c = (1×0−4×6, 4×5−0×0, 0×6−1×5) = (-24, 20, -5). Then a·(-24,20,-5) = 1(-24)+2(20)+3(-5) = -24+40-15 = 1, so the volume is |1| = 1 — a very thin, nearly flat parallelepiped despite none of the vectors being obviously small.

Common mistakes to avoid

Computing a×(b·c) or another mismatched combination of dot and cross products — the correct order is always a dotted with (b crossed with c).
Forgetting to take the absolute value and reporting a negative "volume" when the vectors happen to form a left-handed set.
Assuming a zero result means an input error, rather than recognizing it correctly indicates the three vectors are coplanar.
Mixing up which three vectors represent edges from the same shared vertex versus vectors representing entire opposite edges of the solid.

Frequently asked questions

The raw scalar triple product is signed — positive or negative depending on whether the three vectors form a right-handed or left-handed set — but a physical volume is always a non-negative quantity, so the sign is discarded.

It means the three vectors are coplanar — they all lie flat in the same plane rather than spanning genuine 3D space — so the "parallelepiped" collapses to zero volume, like a flattened box.

The scalar triple product a·(b×c) is exactly equal to the determinant of the 3×3 matrix whose rows (or columns) are the components of a, b, and c — computing the triple product is one concrete way of evaluating that determinant.

The magnitude (and therefore the reported volume) stays the same for any cyclic reordering of a, b, c, but swapping any two of them flips the sign of the raw triple product — since the calculator takes the absolute value, this has no effect on the final answer.

A tetrahedron formed by the same three edge vectors has exactly one-sixth the volume of the full parallelepiped, since the parallelepiped can be divided into six tetrahedra of equal volume.

Then their cross product is the zero vector (since parallel vectors span no area), which forces the entire scalar triple product to zero — the shape degenerates into a flat parallelogram with no enclosed volume.