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# Volume of Cuboid – Formulas, Derivation & Examples

September 24, 2022

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.

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.

## 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 $l \times w \times h = lbh \text{ } unit^{3}$.

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 $h = 5 m = 500 cm$