A parallelepiped is a polyhedron with six faces, and each face is a parallelogram. Basically, they are formed by six parallelogram sides, which results in forming a 3-D figure or a prism. A special case of a parallelogram can be a cube, cuboid, or rhomboid, depending on the length of the sides. In this article, we derive the formula of a parallelepiped using some basic vector multiplications.
Derivation for the formula for the volume of a parallelopiped
Let us calculate the formula for the volume of a parallelepiped as shown in the figure below.
The sides of the parallelepiped are represented by vectors a, b, and c.
The volume of any quadrilateral can be calculated by multiplying the area of the base by the height. Here the base area will be the cross-product of the two vectors b and c. The direction of b x c will be perpendicular to the basal plane. The height will be |a| |cos Φ|. Thus the volume of the parallelopiped can be calculated as |( b x c ). a|.
The formula for the volume of a parallelepiped
The volume of the parallelepiped,
V = |bxc||c||cos Ï•|
  = |( b x c ). a|
Difference between the volume of parallelopiped and parallelogram
A parallelogram is a 2-D quadrilateral, so it does not have a volume. But it has an area with a formula of b*h, where b is the breadth and h is the height of the parallelogram. On the other hand, a parallelopiped is a 3-D polyhedron, so it has a volume and surface area. The surface area of a parallelopiped can be calculated as the sum of all six parallelograms.
Real-life examples of a parallelopiped
- Eraser
- Blocks of stone
- Cubes
- Cuboids
- Rhomboids
See Also
Area of a sector formula
Perimeter of a sector-Formula
Perimeter of a right-angled triangle
Every square is a rectangle
Orthocenter of a triangle
