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Volume of Cuboid – Formulas, Derivation & Examples

volume of cuboid

A cuboid is a 3D shape similar to a cube but has different measurements for length, width, and height. polyhedron surrounded by $6$ rectangular faces with $8$ vertices and $12$ edges is called a cuboid. One of the most common real-world examples of a cuboid is a rectangular box.

The space occupied by a cuboid is called the volume of cuboid. It’s also referred to as the capacity of a cuboid and it depends on the length, width(or breadth), and height of a cuboid. Volume of cuboid is measured in terms of cubic units, such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$, etc.

 Let’s learn how to find the volume of cuboid and its formula.

Cuboid – A 3D Solid Shape

A cuboid is a polyhedron having $6$ rectangular faces, $8$ vertices, and $12$ edges. The opposite faces of the cuboid are parallel and adjacent faces are perpendicular to each other. The length, width(or breadth), and height are of different measurements in a cuboid since the 3D figure is a rectangle that has sides of different lengths.

volume of cuboid

A cuboid has the following properties:

  • A cuboid has $12$ edges, $6$ faces, and $8$ vertices.
  • The faces are shaped as a rectangle hence the length, width(or breadth), and height are different.
  • The angles between any two faces or surfaces are $90^{\circ}$.
  • The opposite planes or faces in a cuboid are parallel to each other.
  • The opposite edges in a cuboid are parallel to each other.
  • Each of the faces in a cuboid meets the other four faces.
  • Each of the vertices in a cuboid meets the three faces and three edges.

Difference Between Cuboid and Cube

Although cuboid and cube are similar $3D$ objects, there are few differences between these two. Following are the differences between a cuboid and a cube.

volume of cuboid
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What is the Volume of  Cuboid?

The volume of cuboid is the number of cubic units, occupied by the cuboid completely. The unit of volume of cuboid is cubic units such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$.

If the measure of length, width(breadth) and height of a cuboid are $l$, $w$ and $h$ unit, then the formula for calculating its volume is $l \times w \times h = lbh \text{ } unit^{3}$.

volume of cuboid

Note: The volume of a cube of the length of each edge $a$ is $a^{3}$. In the case of a cube, $l = w = h = a$, therefore, the volume of a cube is $a \times a \times a = a^{3}$.

Relation Between Volume and Area of Base of a Cuboid

Consider a cuboid with dimensions: length = $l$, width = $w$ and height = $h$

The volume of cuboid $V = l \times w \times h$.

The area of the base (floor) of a cuboid is given by $A = l \times w$.

Replacing $l \times w$ by $A$ in the volume formula, we get $V = A \times h$.

Therefore, $\text{Volume of Cuboid} = \text{Area of Base} \times \text{Height}$.

Examples

Ex 1: Find the volume of cuboid of the length of $20 cm$, the width of $15$ cm, and the height of $10 cm$.

The dimensions of the cuboid are 

$l = 20 cm$, $w = 15 cm$, and $h = 10 cm$

Volume of cuboid = $l \times w \times h = 20 \times 15 \times 10 = 3000 c.c.$.

Note: $c.c.$ means cubic centimetres and is same as $cm^{3}$.

Ex 2: A wall has to be built with a length of $12 m$, a thickness of $80 cm$, and a height of $5 m$. Find the volume of the wall in cubic cm.

The dimensions of the cuboidal wall are

Length $l = 12 m = 1200 cm$

Width $w = 80 cm$

Height $h = 5 m = 500 cm$

The volume of wall = $l \times w \times h = $1200 \times 80 \times 500 = 4,80,00,000 c.c.$.

Ex 3: If the volume of a room is $720 ft^{3}$ and the area of its floor is $120 ft^{2}$, then find the height of the room.

Volume of a room $V = 720 ft^{3}$

Area of room’s floor $A = 120 ft^{2}$

Let the height of the room be $h ft$, then $V = A \times h => 720 = 120 \times h => h = \frac {720}{120} = 6 ft$.

The height of the room is $6 ft$.

Ex 4: The volume of cuboid is $600 cm^{3}$. If its length and width are $25 cm$ and $8 cm$ respectively, find the height of the cuboid.

Volume of cuboid $V = 600 cm^{3}$

Length $l = 25 cm$ and width $w = 8 cm$

Let the height of the cuboid be $h$.

$V = l \times b \times h => 600 = 25 \times 8 \times h =>600 = 200 \times h => h = \frac {600}{200} => h = 3$

Therefore, the height of the cuboid is $3 cm$.

Ex 5: The base of a cuboid is a square. If the height and the volume of cuboid are $4.5 cm$ and $450 c.c.$ respectively, find the length of the base.

The base of a cuboid is a square.

Let the length of the base be $l$ and height be $h$.

$h = 4.5 cm$ and volume $V = 450 c.c.$

$V = l \times l \times h = l^{2} \times h => 450 = l^{2} \times 4.5 => l^{2} = \frac {450}{4.5} => l^{2} = 100$

Taking the square root of both sides

$l = \sqrt{100} => l = 10$

Therefore, the length of the square base is $10 cm$.

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Practice Problems

  1. The length, width, and depth of a water tank are $20 m$, $12 m$, and $4 m$ respectively. Find the capacity of the tank in litres. ($1 m^{3}$ = $1,000$ litre)
  2. The volume of a cuboidal container is $1008 c.c.$. If the length and width of the container are $12 cm$ and $14 cm$ respectively, then find the height of the container.
  3. The dimensions of the brick are $24 cm \times 12 cm \times 8 cm$. How many such bricks will be required to build a wall of $20 m$ in length, $48 cm$ in breadth, and $6 m$ in height? 
  4. The length, width, and height of a cuboid are in the ratio of $3: 5:7$. If the volume of cuboid is $6720 m^{3}$, then find the dimensions of the cuboid.

FAQs

How do you find the volume of cuboid?

The formula for the volume of cuboid is $lwh$, where $l$ is the length, $w$ is the width(breadth), and $h$ is the height of a cuboid.

What is the volume of the cube and cuboid?

The formula for the volume of a cube is $a^{3}$, where $a$ is the length of its edges(sides).
The formula for the volume of cuboid is $lwh$, where $l$, $w$, and $h$ are the length, width, and height of a cuboid respectively.

What is the relation between area and volume?

The area is a region occupied by a 2D figure in a plane, whereas the volume is a space occupied by a 3D object in space. The relation between area and volume is $V = A \times h$, where $V$ is the volume, $A$ is the area of a base and $h$ is the height of an object.

What is the volume cuboid and example?

The volume of a coboid is $\text{length } \times \text{ width } \times { height}$. For example, for a cuboid of length $12 \text{ cm}$, width $8 \text{ cm}$ and height $5 \text{ cm}$, the volume = $12 \times 8 \times 5 = 480 \text{cm}^{3}$.

Conclusion

The volume of cuboid is the number of cubic units, occupied by the cuboid completely and is calculated by finding the product of its length, width(breadth), and height,  i.e., for a cuboid of length $l$, width $w$, and height $h$, its volume is $lwh$. One can also calculate the volume of cuboid if the area of the base and height are known using the formula $Ah$, where $A$ is the area of the base and $h$ is the height of the cuboid.

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