Rectangular Parallelepiped Formula

Parallelepiped Definition

Parallelepiped is a Greek word, which means an object having a parallel plane.

Fundamentally, it is framed by six parallelogram sides to bring about a three-dimensional figure or a Prism, which has a parallelogram base. We can characterize it's anything but a polyhedron, where three sets of parallel faces are combined to frame a three-dimensional shape, having six faces. The cube, cuboid, and rhomboid are its three extraordinary cases. The rectangular parallelepiped has all six faces in a rectangular shape.


So, basically, the formula of rectangular parallelepiped has the three following categories:

  • Rectangular parallelepiped volume formula, and

  • Rectangular parallelepiped surface area formula

On this page, we will understand these two formulas in detail along with illustrative examples.


What is a Rectangular Parallelepiped?

A parallelepiped whose all the six faces possess a rectangular shape, it becomes a rectangular parallelepiped. It is a three-dimensional structure in which the lengths of all the parallel edges are equal to each other.


Rectangular Parallelepiped Definition

A rectangular parallelepiped is a polyhedron with six faces, which is also known as a hexahedron, each of which is a parallelogram.

A hexahedron having three sets of parallel faces, and a prism having a parallelogram base.


Properties of Rectangular Parallelepiped

  • A rectangular parallelepiped is a three-dimensional solid figure.

  • Also, any of its three faces can be viewed at the same time.

  • It comprises three sets of four parallel edges and the edges within each set have equal length measurements.

  • Besides this, the diagonal of each face is called face diagonal.

  • While observing from outside, each face appears to be the mirror image of the opposite face.

  • Its shape indicates a Prism of parallelogram base.

  • Formerly, known as a polyhedron of six faces.

  • Additionally, three pairs of parallel faces form a hexahedron.

Now, let us understand the rectangular parallelepiped formula in detail:


Formula of Rectangular Parallelepiped

From the above text, we understand that Rectangular Parallelepiped is a box shape in three-dimensional space. Formally, it was considered a polyhedron having all faces rectangular in shape.


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From the above rectangular parallelepiped shape, we can see that the base of parallelepiped is in the shape of a parallelogram, so the rectangular parallelepiped base area formula is the area of the parallelogram, which is:

Base Area of Rectangular Parallelepiped  =  Length X Height   =  a x b       

Besides this, the lateral area is the product of the perimeter of the base of the above shape and the height of the six parallelograms faced prism, which is given as:

            LSA = Perimeter of the base of the rectangular parallelepiped  x  Height (h)

In addition to the lateral surface area, the total surface area is the sum of the LSA and twice the base area. Mathematically, we can write this statement as:

                     TSA  =  LSA  +  2 * Area of the base

Rectangular Parallelepiped Volume Formula

At the point when all six faces of parallelepiped are in a rectangular shape, it's anything but a rectangular parallelepiped. It's anything but a three-dimensional box-formed design. The length of all the parallel edges here is equivalent. The base of the prism here is in a rectangular shape. A typical example you can find, in real life, is the shoe box, which has a rectangular shape.

Below is the Rectangular Parallelepiped having length ‘l,’ width ‘w,’ and a height ‘h.’ 


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We know that the rectangular volume formula is:

                                    V  =  l* w* h

We notice that the above formula is also the rectangular parallelepiped volume formula.

Similarly, we can write the rectangular parallelepiped volume formula in the following manner:

                                  V  =  S *  H

Here, 

S is the area of the bottom face, and

H = height of the rectangular parallelepiped


Rectangular Parallelepiped Diagonal Formula

So far we understand that with the help of the three dimensions of the rectangular box, we can find the volume of a rectangular parallelepiped. 

Now, assume that its length = a width = b, and height = c, we can write the formula of volume, surface area, and length of the diagonal of the rectangular box as:

                              V  =  a* b* c

Rectangular Parallelepiped Surface Area Formula  =  2ab + 2bc + 2ac

Looking at the diagonal, we obtain the required rectangular parallelepiped diagonal formula as:

                    Diagonal  =  \[\sqrt{a^{2}+b^{2}+c^{2}}\]

Now, let us have a look at a rectangular parallelepiped example problems with solutions:


Rectangular Parallelepiped Calculator

Assume that the base of a parallelepiped has opposite sides measuring 6 inches and 15 inches and the height of the parallelepiped is 8 inches. So, what will be the cost of painting its walls from outside at a cost of INR 1.3 per square inch?

Solution: 

Here,

The perimeter of the base measures 6 inches, and the height is 8 inches, then we will calculate the lateral surface area of a rectangular parallelepiped: 

                LSA = Perimeter of base × height

                LSA = 2* (6 + 15) × 8

                 LSA = 336 sq.inch

Now, the cost of painting its walls from outside will be:

              Lateral surface area (LSA) × cost per square inch

              Cost of painting the walls = 336 × 1.3 = Rs. 436.8/-


Fun Facts on Rectangular Parallelepiped

For parallelepipeds bearings symmetry plane there are the following two cases:

Having Four Rectangular Faces

Having rhombic faces, while talking about the faces, two adjacent faces are equal and the other two edges are the pairs being a mirror image of each other.


Conclusion

So, what is a rectangular parallelepiped? Well! In a specific term, all six faces of a parallelepiped are parallelograms. In this figure, all pairs of opposite sides are equal. 


This means that every parallelepiped is uniquely linked to a tetrahedron and vice versa because any pair of opposite edges of a tetrahedron specifically defines two parallel planes, one through each of its edges.

FAQs (Frequently Asked Questions)

Q1: What is the Area of a Rectangular Parallelepiped?

Ans: The area of a rectangular parallelepiped equals the sum of its lateral surface area and double of its base area.


Where the lateral surface area is the product perimeter of the base and its height.


And, the total surface area or the TSA = LSA + 2 Base Area, which is equal to the product of the perimeter of the base and height + 2 Base area.

Q2:  What is a Rectangular Cube Called?

Ans: A three-dimensional orthotope is called by three names, i.e., a right rectangular prism, a rectangular cuboid, or a rectangular parallelepiped. However, the peculiar case of an n-dimensional orthotope is the one whose all edges have equal length is the n-cube.

Q3: What are Examples of Rectangular Prisms?

Ans: We find various real-life examples of a right rectangular prism or a cuboid around us. A few of the examples are:

  • Books

  • Boxes

  • Buildings

  • Bricks

  • Blackboards

  • Doors

  • Containers

  • Cabinets

  • Mobiles, and 

  • Notebook PCs or laptops.