Parallelepiped
A parallelepiped is a threedimensional shape that is formed by six parallelograms. The word 'parallelepiped' is derived from the Greek word parallelepipdon, meaning "a body having parallel bodies". We can say that a parallelepiped relates with a parallelogram just like a cube relates with a square. Parallelepiped has 6 parallelogramshaped faces, 8 vertices, and 12 edges. Let us understand properties and different formulas associated with a surface area and volume of a parallelepiped in the following sections.
1.  What Is a Parallelepiped? 
2.  Properties of Parallelepiped 
3.  Surface Area of Parallelepiped 
4.  Volume of Parallelepiped 
5.  Solved Examples 
6.  Practice Questions 
7.  FAQs on Parallelepiped 
What Is a Parallelepiped?
A parallelepiped is a threedimensional shape with six faces, that are all in the shape of a parallelogram. It has 6 faces, 8 vertices, and 12 edges. Cube, cuboid, and rhomboid are all special cases of a parallelepiped. A cube is a parallelepiped whose all sides are squareshaped. Similarly, a cuboid and a rhomboid are parallelepipeds with rectangle and rhombusshaped faces respectively. In the figure given below, we can observe a parallelepiped, with 'a', 'b', and 'c' as side lengths and 'h' as the height of the parallelepiped.
Properties of Parallelepiped
There are certain properties of a parallelepiped that help us distinguish it from other 3D shapes. These properties are listed below,
 Parallelepiped is a threedimensional solid shape.
 It has 6 faces, 12 edges, and 8 vertices.
 All faces of a parallelepiped are in the shape of a parallelogram.
 A parallelepiped has 2 diagonals on each face, called the face diagonals. It has a total of 12 face diagonals.
 The diagonals connecting the vertices not lying on the same face are called the body or space diagonal of a parallelepiped.
 Parallelepiped is referred to as a prism with a parallelogramshaped base.
 Each face of a parallelepiped is a mirror image of the opposite face.
Surface Area of Parallelepiped
The surface area of a parallelepiped is defined as the total area covered by all the surfaces of a parallelepiped. The surface area of a parallelepiped is expressed in square units, like in^{2}, cm^{2}, m^{2}, ft^{2}, yd^{2}, etc. The surface area of parallelepiped can be of two types:
 Lateral Surface Area
 Total Surface Area
Lateral Surface Area of Parallelepiped
The lateral surface area of a parallelepiped is defined as the area of the lateral or side faces of a parallelepiped. To calculate the LSA of a parallelepiped, we need to find the sum of the area covered by the 4 side faces.
Total Surface Area of Parallelepiped
The total surface area of a parallelepiped is defined as the area of all the faces of a parallelepiped. To calculate the TSA of a parallelepiped, we need to find the sum of the area covered by the 6 faces.
Surface Area of Parallelepiped Formula
The formula to calculate the lateral surface area and total surface area of parallelepiped is given as,
LSA of Parallelepiped = P × H
TSA of Parallelepiped = LSA + 2 × B = (P × H) + (2 × B)
where,
 B = Base area
 H = Height of parallelepiped
 P = Perimeter of base
Volume of Parallelepiped
The volume of a parallelepiped is defined as the space occupied by the shape in a threedimensional plane. The volume of a parallelepiped is expressed in cubic units, like in^{3}, cm^{3}, m^{3}, ft^{3}, yd^{3}, etc.
Volume of Parallelepiped Formula
Volume of parallelepiped can be calculated using the base area and the height. The formula to calculate the volume of a parallelepiped is given as,
V = B × H
where,
 B = Base area
 H = Height of parallelepiped
Solved Examples on Parallelepiped

Example 1: If the base face of a parallelepiped has opposite sides measuring 6 inches and 10 inches and its height is 7 inches, find the lateral surface area of the parallelepiped.
Solution:
Using the lateral area of parallelepiped formula,
LSA = Perimeter of base × height
⇒ LSA = 2(6 + 10) × 7
= 224 in^{3}Answer: Lateral area of given parallelepiped = 224 in^{3.}

Example 2: A gift is packed in a rectangular box of dimensions 10 in, 7 in, and 8 in and it needs to be wrapped with gift paper. How much gift paper is required to wrap the gift box?
Solution:
The dimensions of the given gift box are,
length, l = 10 in
width, w = 7 in
height, h = 8 inTo find the amount of gift paper required, we need to find the total surface area of the box. Since the shape of the box can be compared to a rectangular parallelepiped,
TSA = 2 (lw + wh + hl)
= 2 (10 × 7 + 7 × 8 + 8 × 10)
= 2 (70 + 56 + 80)
= 412 in^{2}.
Answer: The amount area of the gift paper required = 412 in^{2}.
FAQs on Parallelepiped
What Is Meant By a Parallelepiped?
Parallelepiped is a threedimensional shape with 6 parallelogramshaped faces, 12 edges, and 8 vertices. Parallelepiped is often referred to as a prism with a parallelogramshaped base. Cube, cuboid, and rhomboid are all special cases of a parallelepiped with faces of the shape of a square, rectangle, and rhombus respectively.
What Is the Volume of a Parallelepiped?
The volume of a parallelepiped is the capacity or the shape or the total space occupied in a threedimensional plane. The volume of the parallelepiped by cubic units, like in^{3}, cm^{3}, ft^{3}, in^{3}, etc.
What Is the Total Surface Area of a Parallelepiped?
The total surface area of a parallelepiped is the area covered by all the faces of a parallelepiped. It is expressed in square units, like in^{2}, m^{2}, cm^{2}, ft^{2}, etc.
What Is the Lateral Surface Area of a Parallelepiped?
The lateral surface area of a parallelepiped is the area or region covered by all the lateral or side faces of a parallelepiped. It is expressed in square units, using units like square inches, square meters, square feet, etc.
What Are the Parallelepiped Formulas?
The formulas associated with a parallelepiped are given as,
 LSA of parallelepiped = P × H
 TSA of parallelepiped = (P × H) + (2 × B)
 Volume of parallelepiped = B × H
where, B is the base area, H is the height of the parallelepiped, and P is the perimeter of base.
What Is a Rectangular Parallelepiped?
A rectangular parallelepiped is a type of parallelepiped whose all six faces are in a rectangular shape and the length of the parallel edges are equal.
What Is the Shape of a Parallelepiped?
Parallelopiped is a 3D shape that has all the sides in the shape of a parallelogram. The opposite faces of a parallelepiped are mirror images of eachother.
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