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**Volume of Parallelepiped Formula**

**Introduction to Volume of Parallelepiped Formula**

A parallelepiped is a three-dimensional 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 parallelogram-shaped 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.

**What Is a Parallelepiped?**

A parallelepiped is a** three-dimensional 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 square-shaped. Similarly, a cuboid and a rhomboid are parallelepipeds with rectangle and rhombus-shaped 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 3-D shapes. These properties are listed below,

- Parallelepiped is a three-dimensional 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 parallelogram-shaped base.
- Each face of a parallelepiped is a mirror image of the opposite face.

**Volume of Parallelepiped**

The volume of a parallelepiped is defined as the space occupied by the shape in a three-dimensional plane. The volume of a parallelepiped is expressed in cubic units, like in3, cm3, m3, ft3, yd3, 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 Volume of Parallelepiped Formula**

**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 in2

Lateral area of given parallelepiped = 224 in2.

**Ex****ample 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 in

To 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 in2.

The amount area of the gift paper required = 412 in2.

**Frequently Asked Questions on Volume of Parallelepiped Formula**

#### What Is Meant By a Parallelepiped?

Parallelepiped is a three-dimensional shape with 6 parallelogram-shaped faces, 12 edges, and 8 vertices. Parallelepiped is often referred to as a prism with a parallelogram-shaped 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 three-dimensional plane. The volume of the parallelepiped by cubic units, like in3, cm3, ft3, in3, 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 in2, m2, cm2, ft2, 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 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 3-D shape that has all the sides in the shape of a parallelogram. The opposite faces of a parallelepiped are mirror images of eachother.

#### How do you find the base area of a parallelepiped?

The three pairs of parallel faces form a hexahedron. For a given parallelepiped, let S is the area of the bottom face and H is the height, then the volume formula is given by; Since the base of parallelepiped is in the shape of a parallelogram, therefore we can use the formula for the area of the parallelogram to find the base area.

What Are the Parallelepiped Formulas?

Answer: 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.