Volume of Cuboid – Definition, Formula, Derivation, Examples, FAQs

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What Is the Volume of a Cuboid?

The volume of a cuboid is the amount of space occupied by the cuboid. It is calculated by multiplying the length, width, and height of the cuboid.

Dimensions of a cuboid

In geometry, a cuboid is a geometric solid with 6 faces, 12 edges, and 8 vertices.. The opposite faces of every cuboid are equal.

Faces, edges, and vertices of a cuboid

What do you mean by the volume of a cuboid? Just like area denotes the space occupied by an object on a 2D plane, the volume represents the space occupied by a solid in the 3D space. A cuboid is a three-dimensional solid. The total 3D space occupied by a cuboid is its volume.

A common real-life example of the volume of a cuboid is the amount of water that completely fills a cuboid-shaped aquarium.

Volume of a Cuboid Formula

The volume of cuboid is a product of its length, breadth, and height. 

The volume of a cuboid formula is written as

Volume of a cuboid $=$ length $×$ breadth $×$ height

Volume of a cuboid $= l \times b \times h$    

Volume is measured in cubic units.

Volume of a cuboid formula

How to Calculate the Volume of a Cuboid

Step 1: Note down the dimensions of the given cuboid as $l = length,\; b = breadth,$ and $h = height$. 

Step 2: Check whether they are all in the same unit or not. If we come across length, breadth, or height in different units, convert them into the same unit. 

Step 3: Substitute the values l, b, and h in the volume formula $V = l \times b \times h$. 

The resultant value will be the volume of a cuboid. It will be written with cubic units.

Volume of a Cuboid Prism

Cuboid prism or rectangular prism are just other names for a cuboid. A cuboid prism has a rectangular cross-section. It is called a right prism when the angles between its sides (lateral faces) and the base are right angles. Its top surface and the corresponding bottom will be identical. 

A rectangular prism/cuboid prism/cuboid

Therefore, using the same volume of cuboid formula, we can calculate the volume of a cuboid prism:

Volume of a cuboid prism $= l \times b \times h$   (cubic units)

If $l = b = h$, a cuboid becomes a cube. Its volume is given by $(side)^{3}$.

Volume of cube and cuboid are both expressed in $unit^{3}$.

Derivation of the Volume of a Cuboid

Interestingly, we can also calculate the volume of cuboid if we know its base area and height. Suppose the area of the cuboid’s rectangular face is A and the height of the cuboid is “h.” 

Since volume is the space occupied, the mathematical expression will be as follows 

Volume of a rectangular prism $=$ Base area $\times$ Height

$V = A \times h$ ————— (i)

As we know that the area of a rectangular surface can be calculated using the following formula:

$Area = length \times breadth$   

$A = l \times b$ —————- (ii)

Substituting equation (ii) in equation (i), we get the following:

$V = A \times h$

$V = (l \times b) \times h$

Thus, we get the formula of a cuboid as follows:

$V = l \times b \times h$

Volume of Cuboid Using Unit Cubes

A unit cube is a cube whose each side is 1 unit. The volume of a cuboid can also be defined as the number of unit cubes that fit perfectly into the cuboid.

A cuboid composed of 8 unit cubes

You can see that 8 unit cubes perfectly fit into the given cuboid.

Thus, volume of cuboid $= 8$ cubic units

Facts about Volume of Cuboid

  • Cuboid is also known as a rectangular prism, rectangular box, rectangular parallelepiped, or a rectangular brick!
  • Total Surface Area of Cuboid $= 2(lb + bh + hl)$

Conclusion

In this article, we learned how to find the volume of a cuboid, its formula, derivation, and examples. Now, we will solve a few examples and practice problems for revision.

Solved Examples of Volume of a Cuboid

1. What is the volume of the given cuboid?

Cuboid with dimensions 6 inches, 4 inches, and 2 inches

Solution: 

$l = 6$ inches

$b = 2$ inches

$h = 4$ inches

Volume of cuboid $= l \times b \times h$

Volume of cuboid $= 6 \times 4 \times 2$

Volume of cuboid $= 48\; inches^{3}$

2. The dimensions of the cuboid-shaped aquarium are: $h = 5$ inches, $l = 10$ inches, and $b = 8$ inches. What is the volume of the cuboid?

Cuboidal aquarium

Solution: 

$l = 10$ inches

$b = 8$ inches

$h = 5$ inches

Volume of cuboid $= l \times b \times h$

Volume of cuboid $= 10 \times 8 \times 5$

Volume of cuboid $= 400\; inches^{3}$

3. Find the length of the cuboid, if its volume is $24\; inches^{3}$. Given: breadth $= 6$ inches and height $= 2$ inches.

Solution: 

Let the length of the cuboid be l.

Volume of cuboid $= l \times b \times h$

$24 = l \times 6 \times 2$

$24 = l \times 12$

$2412 = l$

$l = 2$ in

Therefore, the length of the cuboid is 2 inches.

Practice Problems for Volume of a Cuboid

Volume of Cuboid - Definition, Formula, Derivation, Examples, FAQs

Attend this quiz & Test your knowledge.

1

What is the volume of the given cuboid?

Volume of Cuboid – Definition, Formula, Derivation, Examples, FAQs
$50\; feet^{3}$
$100\; feet^{3}$
$70\; feet^{3}$
$20\; feet^{3}$
CorrectIncorrect
Correct answer is: $100\; feet^{3}$
Using the volume of cuboid formula, we will find the product of 10, 5, and 2 feet.
$V = l \times b \times h = 10 \times 2 \times 5 = 100 feet^{3}$.
2

A cuboid-shaped swimming pool is 20 feet long, 10 feet deep, and 8 feet wide. What is the volume of the swimming pool?

$1600\; feet^{3}$
$600\; feet^{3}$
$800\; feet^{3}$
$1000\; feet^{3}$
CorrectIncorrect
Correct answer is: $1600\; feet^{3}$
Using the volume of cuboid formula, we will find the product of 20, 10, and 8 feet.
$V = l \times b \times h = 20 \times 10 \times 8 = 1600\; feet^{3}$.
3

The formula to find the volume of a cuboid with length l, breadth b, and height h is ___

$\frac{1}{2} \times l \times b \times h$
$lb + bh + lh$
$l \times b \times h$
hl
CorrectIncorrect
Correct answer is: $l \times b \times h$
The volume of the cuboid formula is $l \times b \times h$.
4

What is the volume of the given rectangular prism?

Volume of Cuboid – Definition, Formula, Derivation, Examples, FAQs
$80\; feet^{3}$
$10\; feet^{3}$
$70\; feet^{3}$
$16\; feet^{3}$
CorrectIncorrect
Correct answer is: $80\; feet^{3}$
Using the volume of cuboid formula, we will find the product of 8, 2, and 5 feet.
$V = l \times b \times h = 8 \times 2 \times 5 = 80\; feet^{3}$.

Frequently Asked Questions about Volume of Cuboid

Since all sides of a cube are the same, the volume of a cube is equal to the cube of its side. Mathematically, $V = side^{3}$.

To convert a given volume in $inch^{3}$, we will divide the volume value by 1728 to get its $feet^{3}$ equivalent.

As the volume of cuboid $= length \times width \times height$, we will double the length. 

So, $2l \times b \times h = 2$ volume. 

Thus, the volume of the cuboid will be doubled when we double the length of its side.