Volume of a Parallelepiped Calculator
Enter the three edge vectors, the edge lengths with angles, or four corner vertices to find the volume and surface area of a parallelepiped instantly. Switch between the three input modes to match how your problem is stated. The step-by-step panel shows every calculation with your actual numbers.
Formula
Worked example
With vectors a = (2, 0, 0), b = (0, 3, 0), c = (0, 0, 4): b x c = (3*4-0, 0-0, 0-0) = (12, 0, 0); a . (12, 0, 0) = 24. Volume = |24| = 24 cubic units. This is the same as a 2x3x4 cuboid, as expected.
What is a parallelepiped?
A parallelepiped is a three-dimensional figure formed by six parallelogram faces. Think of it as the three-dimensional generalization of a parallelogram: just as a parallelogram is defined by two edge vectors in a plane, a parallelepiped is defined by three edge vectors in space. When all angles between adjacent edges are 90 degrees, the result is a rectangular parallelepiped, more commonly called a cuboid or box. The word parallelepiped comes from the Greek for "parallel plane surface." It arises naturally in linear algebra (as the geometric meaning of the determinant), crystallography (crystal unit cells are parallelepipeds), and physics (flux calculations).
Three ways to calculate the volume
Method 1 - Scalar triple product (vector form): if the three edge vectors from one vertex are a, b, and c, the volume is the absolute value of the scalar triple product V = |a . (b x c)|. This equals the absolute value of the 3x3 determinant whose rows are the component vectors. Method 2 - Edge lengths and angles: given edge lengths a, b, c and the angles alpha (between b and c), beta (between a and c), and gamma (between a and b), the volume is V = a*b*c * sqrt(1 + 2*cos(alpha)*cos(beta)*cos(gamma) - cos^2(alpha) - cos^2(beta) - cos^2(gamma)). When all three angles equal 90 degrees, the formula collapses to V = a*b*c, the familiar box formula. Method 3 - Four vertices: knowing the coordinates of vertex P and its three neighbours Q, R, S, form edge vectors PQ, PR, PS and apply the scalar triple product.
Surface area of a parallelepiped
A parallelepiped has six faces that come in three pairs of congruent parallelograms. The area of each parallelogram is the magnitude of the cross product of its two edge vectors. The total surface area is A = 2 * (|a x b| + |b x c| + |a x c|). In the edge-and-angle form this becomes A = 2 * (a*b*sin(gamma) + b*c*sin(alpha) + a*c*sin(beta)), where gamma is the angle between a and b, and so on. For a cuboid all three terms simplify to a*b, b*c, and a*c giving the familiar A = 2(ab + bc + ac).
The scalar triple product and the determinant
The scalar triple product a . (b x c) equals the determinant of the 3x3 matrix whose rows (or columns) are the vectors a, b, and c. The sign of the determinant tells you the orientation: a positive value means a, b, c form a right-handed system (the same convention as the standard x, y, z axes), and a negative value means a left-handed system. Volume is always non-negative, so we take the absolute value. In linear algebra, this determinant is the scaling factor by which the linear transformation defined by the matrix changes volumes: a matrix with determinant 2 doubles the volume of any region it maps. This connection makes the parallelepiped volume formula central to multivariable calculus, particularly when changing variables in triple integrals via the Jacobian.
Parallelepiped special cases
| Special case | Angle conditions | Volume formula | Shape |
|---|---|---|---|
| Rectangular parallelepiped (cuboid) | alpha = beta = gamma = 90 deg | V = a * b * c | All faces are rectangles |
| Oblique with one right pair | One angle = 90 deg | V = a*b*c * sqrt(1 - cos^2(other angles) ...) | Four rectangular + two oblique faces |
| Rhombohedron | a = b = c, alpha = beta = gamma | V = a^3 * sqrt(1 + 2cos^3(a) - 3cos^2(a)) | Six congruent rhombus faces |
| Cube | a = b = c, alpha = beta = gamma = 90 deg | V = a^3 | All faces are squares |
How the general parallelepiped simplifies under special angle conditions.
Frequently asked questions
What is the formula for the volume of a parallelepiped?
The most general formula is V = |a . (b x c)|, the absolute value of the scalar triple product of the three edge vectors. If you know only edge lengths and angles, use V = a*b*c * sqrt(1 + 2*cos(alpha)*cos(beta)*cos(gamma) - cos^2(alpha) - cos^2(beta) - cos^2(gamma)). For a rectangular parallelepiped (cuboid) the formula simplifies to V = length * width * height.
How is a parallelepiped different from a cuboid?
A cuboid is a special parallelepiped in which all three pairs of adjacent edges meet at right angles (90 degrees), so all six faces are rectangles. In a general parallelepiped the faces are parallelograms and the angles between edges can be anything between 0 and 180 degrees (exclusive). Every cuboid is a parallelepiped, but not every parallelepiped is a cuboid.
Can the scalar triple product be negative?
Yes. The scalar triple product a . (b x c) is negative when the vectors a, b, c form a left-handed system rather than a right-handed one. This happens, for example, if you swap any two of the vectors. Volume is always non-negative, so we report |a . (b x c)|. The calculator shows you both the signed triple product and the unsigned volume.
What is the perpendicular height of a parallelepiped?
The perpendicular height is the distance between the two opposite faces that you choose as the base. It equals the Volume divided by the base face area. For the base spanned by a and b, height h = |a . (b x c)| / |a x b|. This is analogous to a parallelogram where area = base * height: for a parallelepiped, V = base area * perpendicular height.
How do I find the volume from four vertices?
Label one vertex P and call the three adjacent vertices Q, R, and S. Form the edge vectors a = Q - P, b = R - P, and c = S - P (each is just component-by-component subtraction). Then compute the scalar triple product V = |a . (b x c)|. This calculator does all of that for you: enter the coordinates and switch to the "four vertices" mode.
What happens to the volume if I double all edge lengths?
Doubling all three edge lengths scales the volume by 2^3 = 8. More generally, scaling all edges by a factor k multiplies the volume by k^3 and the surface area by k^2. This cubic scaling is why a basketball holds roughly eight times the air of one half its radius.