Volume of Cuboid: Definition, Formula, Solved Examples

A cuboid is a three-dimensional solid figure that can be found in different things around us such as lunch boxes, matchboxes, books, cupboards, dusters, etc. Volume of Cuboid is the amount of space contained by a cuboid. The unit of volume of a cube will be the cube of any unit of length like \({\text{m}}{{\text{m}}^3},\,{\text{c}}{{\text{m}}^3},\,{{\text{m}}^3},\) etc.

The cuboid’s six faces are arranged in a pair of three parallel faces. The volume of a cuboid is equal to the product of length, breadth and height of a cube as it is a three-dimensional solid figure. The volume of a cuboid depends on the length of its edges. In the below article, we will learn about the volume of the cuboid formula, and how to find it.

What is a Cuboid?

A cuboid is known as a three-dimensional object which has six faces, twelve edges, and eight vertices. All faces of a cuboid are rectangles. All edges are defined as length, breadth, and height. It means all three dimensions of a cuboid are different in length.

What is Volume?

The volume of a three-dimensional solid is the amount of space it occupies or space enclosed by a boundary or occupied by an object or the capacity to hold something.

For example, the volume of a cuboidal box indicates the amount of water or any substance that can be contained in it. In the case of a box full of sand, the amount of sand can be filled inside the box without leaving any gaps in the volume of the box.

What is the Volume of a Cuboid?

The volume of a cuboid is the amount of space contained by it. The volume of a cuboid is measured in cubic units like \( {{\text{m}}^3},\,{\text{c}}{{\text{m}}^3},\,\) etc.

How to Identify the Shape of a Cuboid?

The cuboid has six faces, eight vertices, and twelve edges. All faces of a cuboid are rectangles. Let’s start with the fundamental parameters of three-dimensional objects, such as the face, vertex, and edge.

Faces

All distinct flat surfaces of a solid object are known as the face of that object. A cuboid has six rectangular faces.

Edge

An edge is a line segment joining two vertices.

Vertex

A point where two or more lines meet is known as a vertex. A vertex of a cuboid means a meeting point of two edges. Also, a corner can be called a vertex. A cuboid has twelve edges so, it has eight vertices.

Types of Cuboid

There are two types of Cuboid:

Solid Cuboid: A solid cuboid is a three-dimensional object that is in the form of a cuboid and filled up with the material it is made up of. For example, we can consider bricks.

Hollow Cuboid: A hollow cuboid is a cuboid that has only the outer cubical wrapper and nothing is filled inside.

For easy understanding, consider a cuboid-shaped cardboard box.

What Do We Understand by Volume of a Cuboid?

Let us understand the Volume of Cuboid with an example. Try to fill a cuboid container with sand. Now, pour the water into another vessel and measure the mass of it contained in the vessel. Now, we will get the volume of the cuboid container if we know the density of sand. \({\rm{volume = }}\frac{{{\rm{ mass }}}}{{{\rm{ density }}}}\)

What is Cuboid Formula?

The volume of a cuboid can be calculated when the length, breadth, and height are given. A cuboid is a three-dimensional object like a cube. A cuboid can have a different measure of side lengths and the length of the edges of a cube are the same. We know the volume of any three-dimensional figure \({\rm{area}}\,{\rm{of}}\,{\rm{the}}\,{\rm{base}} \times {\rm{height}}\)

A cuboid has six rectangular surfaces. Let us consider it as the base. The volume of the cuboid \({\rm{ = length \times breadth \times height}}\)

Derivation of Volume of a Cuboid Formula

Let us take a rectangular sheet with its sides of \(l = x\,{\text{unit}},\,y\,{\text{unit}}.\) Now, we will find the area of the rectangular sheet. Therefore, the area of the sheet is \(\left({x \times y} \right)\,{\text{uni}}{{\text{t}}^2}.\) A cuboid can be formed by heaping multiple sheets one upon the other till the height equals \(h = z\,{\text{unit}}.\) Thus, the volume of the cuboid can be derived by calculating the product of the area of the rectangle and the height. So, the volume of a cuboid will be given by \(l \times b \times h = xyz\,{\text{uni}}{{\text{t}}^3}.\)

Unit of Volume of a Cuboid

The unit of length of a cuboid can be any units of length like mm,cm,m, etc. Since the side length is cubed in the formula of volume of a cuboid, the unit should also be cubed. Hence, the unit of volume of a cuboid will be the cube of any unit of length like \({\text{m}}{{\text{m}}^3},\,{\text{c}}{{\text{m}}^3},\,{{\text{m}}^3},\) etc.

Applications

The understanding of the volume of a cuboid helps us in many ways. There are a lot of different things we use in daily life. These involve different kinds of boxes, metal cuboids, etc. All these are cubes but vary in size and texture. The size of these cuboids is decided by the sides of the cuboids.

Steel or metal cuboids are used for many industrial purposes as well. The knowledge of volume is essential in calculating the mass of something. Suppose if we need to find the mass of a metal cuboid, we can find out if we know the density of the material as: \({\rm{mass = volume \times density}}\)

Solved Examples

Q.1. Find the cuboid volume, of length, breadth, and height \(5\,{\text{cm}},\,3\,{\text{cm}},\,10\,{\text{cm}}\) respectively.
Ans : Given, the length of the sides of the cuboid are \(5\,{\text{cm}},\,3\,{\text{cm}},\,10\,{\text{cm}}\).
We know, the volume of a cuboid \({\rm{ = length \times breadth \times height}}\)
Therefore, volume \( = (5 \times 3 \times 10){\rm{c}}{{\rm{m}}^3}\)
\(v = 150\,{\text{c}}{{\text{m}}^3}\)
Hence, the volume of the cuboid is \(= 150\,{\text{c}}{{\text{m}}^3}.\)

Q.2. Find the length of the height of the cuboid if its base area is \( = 120\,{\text{c}}{{\text{m}}^2}\) and the volume is \( = 1800\,{\text{c}}{{\text{m}}^3}.\)
Ans : Given, the volume of the cube \( = 1800\,{\text{c}}{{\text{m}}^3}.\)
Let the length of the height is \(h.\)
We know,\({\rm{ = length \times breadth \times height = area of the base \times height}}\) the volume of a cuboid Substituting the value, we get, \( = 120 \times h = 1800\) \( \Rightarrow h = \frac{{1800}}{{120}} \Rightarrow h = 15\,{\text{cm}}\) Thus, the height of the cuboid is \(15\,{\text{cm}}\)

Q.3. What will be the length of the cuboid if its volume is \(3000\,{\text{c}}{{\text{m}}^3}\), breadth is \(10\,{\text{cm}}\) and height is \(10\,{\text{cm}}\,{\text{?}}\)
Ans: We know, the volume of a cuboid is given as: \({\rm{volume = length \times breadth \times height}}\)
The given dimensions for cuboid are:
Volume \(= 3000\,{\text{c}}{{\text{m}}^3}\)
Breadth \(= 10\,{\text{cm}}\)
Height \(= 5\,{\text{cm}}\)
Let the length of cuboid is \(= h\,{\text{cm}}{\text{.}}\)
Hence, the volume of the cuboid will be: \( \Rightarrow {\rm{volume}} = h \times 10 \times 5 = 3000\)
\( \Rightarrow h = \frac{{3000}}{{50}} = 60\,{\text{cm}}.\) \(\therefore \) The length of cuboid is \(60\,{\text{cm}}.\)

Q.4.  If all the dimensions of a cuboid are doubled, by what factor does its volume change?
Ans: We know, a cube has three dimensions, these are length, breadth, and height.
Let us say, length \( = l\), breadth \( = b\), height \( = h\)
Case 1: The volume of the cuboid \( = l \times b \times h\) Case 2: If the length, breadth, height are doubled then the volume of the cuboid = 2l×2b×2h=8lbh
Hence, if all the dimensions are doubled then, the volume of the cuboid increase \(8\) times.

Q.5: In a cuboid, the length is double of its breadth and its volume is \(40\,{{\text{m}}^3}\) Find its length and breadth if the height is \( {\text{5 m}}{\text{.}}\)

Ans: Let us assume the breadth of the cuboid is \(x\,{\text{m}}\) and the length is \(2x\,{\text{m}}{\text{.}}\)
We know, the volume of the cuboid is \( = {\rm{length \times breadth \times height}} \Rightarrow 2x \times x \times 5 = 40\)
\( \Rightarrow 10{x^2} = 40 \Rightarrow {x^2} = \frac{{40}}{{10}} \Rightarrow x = 2\)
Hence, the length \( = 2x = 4\,{\text{cm}},\) breadth \(= x = 2\,{\text{cm}}\), the length of the cuboid is \(4\,{\text{cm}}\) and breadth is \(2\,{\text{cm}}{\text{.}}\)

Summary

A cuboid is a three-dimensional object which has six rectangle shape faces and twelve edges, and eight vertices. The three dimensions of the cuboid are length, breadth, and height. We can find the volume of a cuboid by calculating the product length, breadth, and height. There are different uses of the cuboid in real life and the volume of a cuboid helps us to find the mass of the cuboid if the density is given. Here, we have discussed how to find the volume of a cuboid using formulae.

FAQs

Q.1. What is the formula of the diagonal of a cuboid?
Ans : Diagonal of a cuboid \(\sqrt {{{({\rm{ length }})}^2} + {{({\rm{ breadth }})}^2} + {{({\rm{ height }})}^2}} \)

Q.2. What is the volume in cubic units of a cuboid?
Ans : We know, the volume of a cuboid is \({\rm{length \times breadth \times height}}{\rm{.}}\)
If all the units of each dimension are m then, the unit of the volume is \({m^3}.\)

Q.3. How do you find the length of a cuboid?
Ans : The volume of the cuboid is \({\rm{length \times breadth \times height}}{\rm{.}}\) If the volume, breadth, height are given then we can write, \(v = l \times b \times h \Rightarrow l = \frac{v}{{bh}}.\)

Q.-4: What is the volume of cube, cuboid and cylinder?
Ans: So, the volume of a cube \( = {({\rm{ side }})^3} = {a^3}\)
The volume of a cuboid \({\rm{ = length \times breadth \times height}} = l \times b \times h\)
The volume of a cylinder \( = \pi {r^2}h\) when \(r\) is the radius and \(h\) is the height of the cylinder.

Q.5. How do we get the volume of a cuboid formula?
Ans : The volume of the cuboid is \({\rm{ = length \times breadth \times height}}\)

Q.6. What is the formula of volume?
Ans: The formula of volume of a three-dimensional geometric shape is the \({\rm{area of the base \times height}}\)

Now that you have detailed information on the volume of a cuboid, we hope you get through it well. If you face any issue regarding the same do let us know about it in the comments section below and we will get back to you soon.