This volume of a parallelepiped calculator will help you calculate the volume of a parallelepiped from its three vectors, four vertices, or edge lengths. Additionally, it will also calculate the area of the parallelepiped. Are you wondering how to find the volume of a parallelepiped formed by three vectors? Do you want to learn the formula for the volume of a parallelepiped with four vertices? Read on to find out the answers to all of these questions and more.
Volume of a parallelepiped formula
A parallelepiped is a polyhedron whose six faces are parallelograms. To describe a parallelepiped, we need its three adjacent sides and their angles, or the three adjacent vectors. Three co-initial vectors in space describe a parallelepiped.
The formula for the volume of a parallelepiped is given by:
V = |(a⃗ ⨯ b⃗) ⋅ c⃗|
where:
- V — Volume of the parallelepiped formed by the three vectors; and
- a⃗, b⃗ and c⃗ — Three vectors that describe the three adjacent (and unique) sides of a parallelepiped.
The vector multiplication above is called a scalar triple product (or a triple product). It involves the cross product of the vectors a⃗ and b⃗, which results in a vector a⃗⨯b⃗ perpendicular to both a⃗ and b⃗. Check out our cross product calculator if you want to learn more about cross products.
🔎 Note that the vector magnitude of the resultant, |a⃗⨯b⃗|, is equal to the area of a parallelogram described by these two vectors.
How do I calculate the volume of a parallelepiped?
To calculate the volume of a parallelepiped from three vectors, follow these steps:
- Find the cross product a⃗⨯b⃗ of the first two vectors. This gives a new vector perpendicular to both a⃗ and b⃗.
- Take the dot product of that result with the third vector c⃗. The number you get is the scalar triple product (a⃗⨯b⃗)⋅c⃗.
- Take the absolute value of the scalar triple product. That absolute value is the volume V.
The scalar triple product is exactly the determinant of the 3×3 matrix whose rows are the three vectors. Its absolute value equals the volume because the determinant measures the (signed) volume scaling of the parallelepiped spanned by the vectors.
How do I calculate the volume of a parallelepiped from its sides?
If you know the three edge lengths a, b, c and the three angles between them (α between b and c, β between a and c, γ between a and b), the volume follows the formula:
V = a·b·c · √(1 + 2·cosα·cosβ·cosγ − cos²α − cos²β − cos²γ)
When all three angles are right angles (α = β = γ = 90°), the cosines vanish and the formula collapses to V = a·b·c — the familiar volume of a rectangular box (a rectangular parallelepiped, or cuboid).
If you provide four vertices instead, the calculator first builds three co-initial edge vectors by subtracting the shared vertex from the other three, then applies the scalar-triple-product formula above.
How do I calculate the surface area of a parallelepiped?
A parallelepiped has six faces that form three pairs of congruent parallelograms. The area of each parallelogram is the magnitude of the cross product of the two vectors that span it, so the total surface area is:
S = 2·(|a⃗⨯b⃗| + |b⃗⨯c⃗| + |a⃗⨯c⃗|)
In terms of edge lengths and angles this becomes S = 2·(a·b·sin γ + b·c·sin α + a·c·sin β). For a rectangular box the sines are all 1 and it reduces to S = 2·(a·b + b·c + a·c).
How to use this volume of a parallelepiped calculator – And parallelepiped area calculator
- Choose an input method: three vectors, four vertices, or edge lengths & angles.
- Pick your unit system — metric (mm, cm, m, km) or US/Imperial (in, ft, yd, mi). It only affects how the volume (unit³) and area (unit²) are labelled.
- Enter your data:
- Vectors — the components of a⃗, b⃗ and c⃗.
- Vertices — the coordinates of the shared vertex O and the three adjacent vertices P, Q, R.
- Edges — the three edge lengths a, b, c and the three angles α, β, γ between them.
- Read off the volume, the surface area, the three edge vectors, their lengths and angles, the three face areas, and the scalar triple product — together with a step-by-step solution.
Metric vs. US (Imperial) Units
The parallelepiped is a purely geometric object, so the numbers only depend on the values you type, not on the unit label. The unit selector simply reports the results in the units you picked:
- Metric system: lengths in millimeters (mm), centimeters (cm), meters (m) or kilometers (km); volume in the cubed unit (e.g. m³) and area in the squared unit (e.g. m²).
- US / Imperial system: lengths in inches (in), feet (ft), yards (yd) or miles (mi); volume in the cubed unit (e.g. ft³) and area in the squared unit (e.g. ft²).
FAQs
- What is a parallelepiped?
- A parallelepiped is a three-dimensional figure whose six faces are all parallelograms. A cube, a rectangular box (cuboid), and a rhombohedron are all special cases of the parallelepiped.
- How do I find the volume of a parallelepiped from three vectors?
- Compute the scalar triple product V = |(a⃗ ⨯ b⃗) ⋅ c⃗|. First take the cross product of a⃗ and b⃗, then dot the result with c⃗, and finally take the absolute value.
- What does a zero volume mean?
- If the scalar triple product is zero, the three vectors are coplanar (linearly dependent) — they all lie in a single plane, so the "solid" is flat and encloses no volume.
- Is the volume affected by the order of the vectors?
- No. Swapping two vectors flips the sign of the scalar triple product, but because we take the absolute value, the volume is always the same non-negative number regardless of order.
- What is the difference between a parallelepiped and a cuboid?
- A cuboid (rectangular box) is a parallelepiped in which every angle between adjacent edges is 90°. A general parallelepiped allows oblique angles, so its faces are slanted parallelograms rather than rectangles.