Rectangular Parallelepiped Formula

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Parallelepiped Definition

Parallelepiped is the Greek word, which essentially means the object that has a parallel plane. Principally, the Parallelepiped is framed by the six parallelogram sides which bring about the prism or the 3D figure, and it consists of the parallelogram base. It can be categorized as anything but the polyhedron, where 3 sets of the parallel faces are made to combine for framing a three-dimensional (3D) shape that has six faces. The cube, cuboid, and rhomboid are the three exceptional cases. The Rectangular Parallelepiped consists of six faces in a rectangular shape. 


The following are the different categories related to the formula of Rectangular Parallelepiped: Rectangular Parallelepiped volume formula, and the Rectangular Parallelepiped surface area formula. 


So what essentially is the Rectangular Parallelepiped? It is the Parallelepiped where all six faces have the rectangular shape, and thus assumes the shape of a Rectangular Parallelepiped. It consists of a 3D structure, where the lengths of parallel edges tend to be equal to each other. This applies to all the parallel edges. 


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 the 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 of 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 bearing symmetry planes 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.


Facts Associated with the Rectangular Parallelepiped

The Parallelepiped that has symmetry planes have these two cases: Having the rhombic faces, while speaking about faces, the two adjacent faces tend to be equal. The other two edges are pairs and provide the mirror image of each other. To summarize the Rectangular Parallelepiped in the specific form, have six faces as parallelograms. This implies that every Parallelepiped has a unique link to the tetrahedron and also vice versa. This is because any pair having opposite edges of the tetrahedron specifically describes the two parallel planes, one via each of the edges. 


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.

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FAQs (Frequently Asked Questions)

1. What is the area of the Rectangular Parallelepiped?

The area of the Rectangular Parallelepiped is equal to the sum of the lateral surface area and it is twice its base area where the lateral surface area is the product perimeter of the base and the height. Beyond this the total surface area, also referred to as TSA, is equal to LSA plus 2 base areas. This equals the product of the perimeter of the base and the height plus 2 base areas. 

2. What is the rectangular cube known as?

The 3D orthotope is known by 3 names - rectangular cuboid, Rectangular Parallelepiped, or right rectangular prism. The peculiar case however is of one of the n-dimensional orthotopic where all edges having an equal length are determined as then cube. This is the standard definition and explanation of the rectangular cube in the formulae of geometric figures and students need to understand the explanation so they can attempt the questions related to it.

3. What are the different instances of the rectangular prisms?

There are several different real-life instances or examples of the cuboid or the right rectangular prism around us. Some of these examples include Buildings, Books, Boxes, Blackboards, Bricks, Containers, Doors, Mobiles, Laptops, Notebook PCs, and Cabinets, amongst others. These are some of the excellent real-life instances or examples of the rectangular prisms that exist around us and can help students in gaining a better understanding of the concept that they are studying. 

4. What are the features of the Vedantu notes on “Rectangular Parallelepiped formula”?

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