This calculator finds the volume of a parallelepiped in two modes: Rectangular (L × W × H) and Oblique (vectors a, b, c via |a · (b × c)|). A 3D diagram scales to your inputs so you can visualise your figure.
Parallelepiped Volume Calculator
Compute volume as L×W×H (rectangular) or |a·(b×c)| (oblique). Switch mode below.
3D visualization
Length (L)
Width (W)
Height (H)
Vector a (x, y, z)
x
y
z
Vector b (x, y, z)
x
y
z
Vector c (x, y, z)
x
y
z
Volume (V)
Calculation process will appear here.
Decimal Places
Main features
- Two modes matching the toggle: Rectangular and Oblique.
- Configurable decimal places (default 2).
- Copy result to clipboard.
- Live 3D visualization with labeled axes/vectors.
How to use
- Switch the toggle to Rectang. or Oblique.
- Enter:
- Rectang.: Length (L), Width (W), Height (H).
- Oblique: a = (ax, ay, az), b = (bx, by, bz), c = (cx, cy, cz).
- Click Calculate. The result (V) and the diagram update.
- Optionally adjust decimal places or copy the result.
Units & inputs
Use any linear unit (m, cm, in, …). The output unit is cubic (m³, cm³, in³). Zero dimension or coplanar vectors produce V = 0 (degenerate).
Formulas
Rectang.: V = L × W × H.
Oblique: V = |a · (b × c)|, with b × c = (by·cz − bz·cy, bz·cx − bx·cz, bx·cy − by·cx) and a · (b × c) = ax(b×c)x + ay(b×c)y + az(b×c)z.
Determinant form: V = | det([a b c]) | = | ax bx cx |, | ay by cy |, | az bz cz |.
Sources: Wolfram Mathworld, Wiki.
Reference values — Rectang. (examples)
| L | W | H | V |
| 1 | 1 | 1 | 1 |
| 2 | 1.5 | 0.5 | 1.5 |
| 3 | 2 | 1.5 | 9 |
| 4 | 1.2 | 2.3 | 11.04 |
| 5 | 0.8 | 0.4 | 1.6 |
| 2.5 | 2.5 | 2.5 | 15.625 |
| 6 | 3 | 1 | 18 |
| 7.5 | 2 | 0.75 | 11.25 |
| 10 | 1.2 | 0.6 | 7.2 |
| 0.5 | 0.5 | 3 | 0.75 |
| 8 | 4 | 2 | 64 |
| 3.3 | 1.1 | 2.2 | 7.986 |
Reference values — Oblique (examples)
| a | b | c | V = |a·(b×c)| |
| (1, 0, 0) | (0, 1, 0) | (0, 0, 1) | 1 |
| (2, 0, 0) | (0, 3, 0) | (0, 0, 1.5) | 9 |
| (1, 2, 0) | (0, 1, 1) | (1, 0, 1) | 3 |
| (2, 1, 1) | (1, 3, 0) | (0, 2, 4) | 22 |
| (3, −1, 0.5) | (0.5, 2, 1) | (1, 0, 2) | 11 |
| (1, 1, 1) | (1, 2, 3) | (2, 1, 0) | 0 |
| (4, 0, 0) | (1, 1, 0) | (0, 1, 1) | 4 |
| (0, 2, 1) | (1, 0, 2) | (2, 1, 0) | 9 |
| (2.5, 0, 0) | (0, 2.5, 0) | (0, 0, 2.5) | 15.625 |
| (1, 1, 0) | (2, 2, 0) | (0, 0, 5) | 0 |
Notes that matter
- The sign of a·(b×c) is orientation; volume uses |·|.
- a, b, c coplanar ⇒ V = 0 (linearly dependent).
- V² equals det(Gram(a,b,c)).
- Rectang. is the special case a ⟂ b ⟂ c with |a|=L, |b|=W, |c|=H.
- Swapping any two vectors flips the triple-product sign, not |V|.
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