Parallelepiped: Overview, Questions, Preparation

Any three-dimensional figure where all six faces are parallelograms is known as a parallelepiped. They are also referred to as prisms with three pairs of parallel faces. The three cases of parallelepiped are: 

Cuboid:- A prism with a parallelogram base
Rhomboid:- A hexahedron with three sets of parallel faces  
Cube:- A polyhedron with six parallelogram faces.

Characteristics of a Parallelepiped

Here are some of the key characteristics of a parallelepiped:
a. It is possible to see any three faces at a given moment.
b. It is a 3-D solid figure.
c. It has three pairs of 4 parallel edges.
d. All the edges are equal in length.
e.There is a face diagonal for each face of the parallelepiped.

Volume of Parallelepiped
Let us consider a parallelepiped where the base area is ‘B’ and height is ‘H.’ The volume of the parallelepiped is given by:
V = B x H.

In vector form, the volume of a parallelepiped is calculated considering three different dimensions a,b, and c. The formula is:
V = I (a x b).c I

Surface Area of Parallelepiped 

The total surface area of the parallelepiped is determined from the lateral surface area of the parallelepiped. The lateral surface area of a parallelepiped is the product of the height of the parallelepiped faces and the base’s perimeter. Therefore, the formula for calculating the lateral surface area(LSA) of a parallelepiped is: 

LSA = Height x Base Perimeter.

Thus, the total surface area(TSA) is nothing but the arithmetic sum of the lateral surface area and the top and bottom bases of a parallelepiped. Therefore, the formula for calculating TSA is:

TSA = Lateral surface area + 2 x Area of the Base.

The topic parallelepiped is from the chapter titled Vector Algebra. The chapter presents other crucial concepts of scalar dot products and cross products, addition and multiplication of vectors, and scalar triple products of vectors. The chapter accounts for a valuable 14 marks in the final examinations of class 12. Moreover, this topic is essential in national level entrance examinations.

1. Given edges of a parallelepiped as = and  . Find the volume of the parallelepiped.
Solution.
Volume of the parallelepiped = 2[(2 x 2) - (-1 x -1)] - ( -3)[(1 x 2) - (2 x -1)] + 4[(1 x -1) - (2 x 2)]
= 2[4 -1] + 3(2+2) + 4[-1-4]
= 2(3) + 3(4) + 4(-5)
= 6 + 12 - 20
= -2
Therefore, the volume of the parallelepiped is 2 cubic units.

2. A 5-inch long parallelepiped has a top and base measuring 10 inches and 15 inches, respectively. When the cost of painting the box is priced at INR 2 per sq. inch, find the total cost of painting?
Solution.
The total cost of painting can be figured out only with the lateral surface area.

Therefore, LSA = Height x base perimeter
LSA = 2 (10 + 15) x 5
LSA = 250 sq.inch

Thus, the total cost of painting the box = 250 x 2 = Rs. 500/-

3. Composition is 40 cu ft. How many tons would a coal bucket have to have 18 ft. high, 7 ft. wide and 10 ft. deep if maximum level?
Solution.
Now,
The volume of coal bucket can be calculated as:
V = 18x7x10
V = 1260 ft³
No. of tons required is:
W = 1 ton/ 40 ft3(1260 ft³)
W = 31.5 tons

Q: What is the two-dimensional form of a parallelepiped solid figure?

Q: What is the name of a three-dimensional rhombus?

Q: The shape of the parallelepiped resembles which 2-D shape?

Q: When is a parallelepiped referred to as a rectangular parallelepiped?

Q: What is the formula of the face and body diagonal of a parallelepiped?