Volume of a Parallelepiped Calculator
Free Volume aparallelepiped Calculator for coordinate geometry. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
V = |A . (B x C)|
Where A, B, and C are the three edge vectors emanating from a common vertex, B x C is the cross product giving the normal to one face with magnitude equal to that face area, and A . (B x C) is the scalar triple product whose absolute value gives the volume.
Worked Examples
Example 1: Unit Cube Verification
Problem:Verify the volume of a unit cube using vectors A = (1,0,0), B = (0,1,0), C = (0,0,1).
Solution:Cross product B x C = (1*1 - 0*0, 0*0 - 0*1, 0*0 - 1*0) = (1, 0, 0)\nScalar triple product A . (B x C) = 1*1 + 0*0 + 0*0 = 1\nVolume = |1| = 1 cubic unit\nSurface area = 2(|AxB| + |BxC| + |AxC|) = 2(1 + 1 + 1) = 6 square units
Result:Volume = 1 cubic unit | Surface Area = 6 square units
Example 2: Skewed Parallelepiped
Problem:Find the volume of the parallelepiped with edges A = (2, 1, 0), B = (0, 3, 1), C = (1, 0, 2).
Solution:Cross product B x C = (3*2 - 1*0, 1*1 - 0*2, 0*0 - 3*1) = (6, 1, -3)\nScalar triple product A . (B x C) = 2*6 + 1*1 + 0*(-3) = 12 + 1 + 0 = 13\nVolume = |13| = 13 cubic units\n|A x B| = |(1, -2, 6)| = sqrt(41)\n|A x C| = |(2, -4, -1)| = sqrt(21)\nSurface area = 2(sqrt(41) + sqrt(46) + sqrt(21)) = 2(6.403 + 6.782 + 4.583) = 35.54
Result:Volume = 13 cubic units | Surface Area = 35.54 square units
Frequently Asked Questions
What is a parallelepiped and how does it differ from a rectangular box?
A parallelepiped is a three-dimensional figure formed by six parallelograms, where opposite faces are parallel and congruent. Unlike a rectangular box (cuboid), whose faces are all rectangles with right angles, a parallelepiped can have faces that are slanted parallelograms with non-right angles. A rectangular box is actually a special case of a parallelepiped where all angles are 90 degrees. Think of it as taking a cardboard box and pushing it sideways so it leans - the resulting shape is a parallelepiped. Each pair of opposite faces remains parallel, and all edges in the same direction have the same length. This makes the parallelepiped a fundamental shape in crystallography and materials science.
How is the volume of a parallelepiped calculated using the scalar triple product?
The volume of a parallelepiped defined by three edge vectors A, B, and C equals the absolute value of their scalar triple product: V = |A . (B x C)|. First, you compute the cross product of two vectors (B x C), which gives a vector perpendicular to the face they span, with magnitude equal to the area of that parallelogram face. Then you take the dot product of the third vector A with this cross product, which effectively multiplies the face area by the height of the parallelepiped measured perpendicular to that face. The absolute value ensures a positive volume regardless of vector orientation. This elegant formula reduces a complex 3D volume calculation to simple vector arithmetic.
What is the cross product and why is it needed for volume calculations?
The cross product of two vectors A and B produces a new vector that is perpendicular to both A and B, with a magnitude equal to the area of the parallelogram spanned by A and B. The formula is A x B = (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1). In the parallelepiped volume calculation, the cross product of two edge vectors gives both the direction of the normal to one face and the area of that face. The dot product with the third vector then computes the signed height along that normal direction. Without the cross product, computing volumes of skewed 3D shapes would require much more complex trigonometric calculations involving multiple angles between edges.
How do you calculate the surface area of a parallelepiped?
The surface area of a parallelepiped consists of three pairs of parallel parallelogram faces, so the total surface area equals 2 times (|A x B| + |B x C| + |A x C|). Each cross product magnitude gives the area of one parallelogram face. For example, |A x B| is the area of the face formed by edges A and B. Since opposite faces are congruent, you multiply each face area by 2 to get the total. For a rectangular box with edges of lengths a, b, and c, this simplifies to the familiar formula 2(ab + bc + ac). For a general parallelepiped with non-right angles, the cross product automatically accounts for the slant angles, making this formula both general and elegant.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy