How this calculator works
Three adjacent edges form a parallelepiped, a solid with parallelogram faces. The cross product of two edges measures their face area and points perpendicular to it. Dotting the third edge with that cross product gives signed base area times height; its absolute value is the volume. Vertex mode uses PQ, PR and PS, so Q, R and S must each be adjacent to P. Length-angle mode uses the determinant of the pairwise cosine matrix and rejects impossible angle combinations.
Formula
V = |a dot (b cross c)|. S = 2(|a cross b| + |a cross c| + |b cross c|). With edge lengths a,b,c: V = abc sqrt(1 + 2 cos(alpha)cos(beta)cos(gamma) - cos^2(alpha) - cos^2(beta) - cos^2(gamma)).
Worked example
Perpendicular edges (2,0,0), (0,3,0) and (0,0,4) enclose 24 cubic metres. Their distinct face areas are 6, 8 and 12 square metres, so the total surface area is 52 square metres. Translating every vertex leaves these values unchanged.
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Frequently asked questions
Why is the signed triple product sometimes negative?
Its sign records the ordering and orientation of the edges. Exchanging two edges reverses it, while the physical volume stays nonnegative.
Can the volume be zero?
Yes. Coplanar edge vectors give a flattened, degenerate shape with zero volume. The formula still reports the sum of the six face areas, which can overlap in that limit.
Are any three angles valid?
No. Their cosine matrix must be positive semidefinite. A significantly negative determinant is rejected; tiny negative roundoff at a degenerate boundary is clamped to zero.