Volume of cube is one of the basic topics related to the chapter of Mensuration. Cube is a three-dimensional solid figure that can be found in different forms around us such as playing dices, sugar cubes, Rubik’s cube, ice cubes, and so on. The volume of a cube is the amount of space it contains. The volume of a cube is equal to the product of length, breadth, and height of a cube as it is a three-dimensional solid figure. The volume of the cube depends on the length of its edges. All the edges of a cube are equal. The unit of volume of a cube will be the cube of any unit of length like mm3mm3, cm3cm3, m3m3, etc. In this article, let’s discuss the volume of Cube in detail. Read on to find out more.

Embibe offers solution sets for chapter-wise exercises of NCERT books. These solution sets include step wise explanation of each sum which makes it easier for the students to understand the correct approach to answer these questions. The previous year question papers and the mock test papers for class 10th and 12th helps students identify the areas that require further improvements. Students can further access the PDF of the NCERT books on this page itself.

Cube: Definition

A cube is known as a three-dimensional object which has six faces, twelve edges, and eight vertices. All edges of a cube are equal in length. It means all six faces of a cube are squares.

How do we Identify the Shape of a Cube?

The cube has six faces, eight vertices, and twelve edges.
Now, Let’s discuss about basic parameters of cube like face, vertex, and edge which plays a major role in three-dimensional objects.

Faces: All distinct flat surfaces of a solid object are known as the face of that object. A cube has six square faces as all the edges are equal.

Edge: An edge is a line segment joining two vertices.

Vertex: A point where two or more lines meet is known as a vertex. In three-dimensional geometry, this concept is used. A vertex of a cube is the point at which two edges meet. A vertex is another term for a corner. A cube has twelve edges, which means it has eight vertices.

Check Mensuration Formulas Here

What is a Unit Cube?

All the edges of a cube are equal. When each edge of a cube is \(1\,\rm{unit}\) long, it is known as a unit cube.

Cubes: Types

There are two types of Cubes:

1. Solid cube: A solid cube is a three-dimensional object that is in the form of a cube and filled up with the material it is made up of. For example, sugar cubes, ice cubes are solid cubes.

2. Hollow cube: A hollow cube is a cube that has only the outer cubical wrapper and nothing is filled inside. For easy understanding, consider a cardboard box.

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 other substance that can be contained within it.

For example, consider a box full of sand, the amount of sand can be filled inside the box without any gap in the volume of the box. Here is the volume of box will come.

What is the Volume of a Cube?

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

Volume of a Cube – Examples?

Let’s perform an activity to easily understand this. Try to fill an ice tray with water (Ice tray contains twelve cubic moulders). Now, pour the water into a beaker and measure how much water is contained in that. Now, if we divide the result by twelve, we will get the volume of each cubic moulder.

What is the Formula of Volume of a Cube?

There are two methods to compute the volume of a cube.

The volume of a cube when the edge or side is given:

The volume of a cube can be calculated when the edge or side is given. The method is similar to the method of calculating the volume of cuboid. A cuboid is a three-dimensional object which looks like a cube. The length of the edges of a cube are the same. If we know the length of one of the sides of the cube we can easily find out the volume of it. So, the volume of a cuboid \(={\text{length}} \times{\text{breadth}} \times{\text{height}}\)
The volume of a cube \(={\text{side}} \times{\text{side}} \times{\text{side}} = {\left({{\text{side}}}\right)^3}\) \(\left[{\because{\text{length}} = {\text{breadth}} = {\text{height}}}\right]\)

The volume of a cube when the diagonal is given:

As we discussed earlier, a cube has six square faces and all twelve sides of it are equal.
Thus, all possible diagonals are also equal.
Let’s say the side of a cube is \(a\) units and the diagonal is \(d\) units.
Therefore, \( {\text{length}}\, {\text{=}}\, {\text{breadth}}\, {\text{=}}\, {\text{height}}\, {\text{=}}\,a\, {\text{unit}}\).
Since the formula of the diagonal of a three-dimensional object \(=\sqrt{{{\left({{\text{length}}}\right)}^2} + {{\left({{\text{height}}}\right)}^2} + {{\left({{\text{breadth}}} \right)}^2}}\)
We can derive easily the diagonal of a cube from the above formula by substituting length,breadth & height by \(a\,\rm{unit}\).
Hence, \(d = \sqrt{{{\left(a\right)}^2} + {{\left(a\right)}^2} + {{\left(a \right)}^2}} = \sqrt 3 a\)
\( \Rightarrow d = \sqrt 3 a\)
\( \Rightarrow a = \frac{d}{{\sqrt 3 }}\)
The volume of the cube when the diagonal is given \(= \frac{d}{{\sqrt 3 }} \times \frac{d}{{\sqrt 3 }} \times \frac{d}{{\sqrt 3 }} = \frac{{{d^3}}}{{3\sqrt 3 }}\).

How do we Find the Derivation of Volume of a Cube Formula?

Let’s take a square paper with its sides of \(a\,unit\). Now, we will find the area of the square paper.
Therefore, the area of the paper is \({a^2}~{\text{uni}}{{\text{t}}^2}\). Now a cube can be formed by heaping multiple sheets one upon the other till the height equals \(a\,\rm{unit}\).

Thus, the volume of the cube can be derived by calculating the product of the area of the square base and the height of the heap.

So, the volume of a cube will be given by \({a^2} \times a = {a^3}~{\text{uni}}{{\text{t}}^3}.\)

What are the Factors Affecting the Volume of a Cube?

The volume of the cube is determined by the size of the cube. The size is based on the length of the side of the cube. The more the length of the side, the more will be the volume of a cube.

The volume of a cube is given by \(V= {a^3}\), where \(a\) is the side of the cube.
The volume of a cube directly depends on the side length of the cube:
\(V\propto a\)

If we compare two cubes of different sizes, the cube with greater side length will be having a larger volume.

Here, the red cube is having a side length of \(2\,\rm{m}\) and the yellow cube is having a length of \(3\,\rm{m}\). As the yellow cube is having a greater side length, the volume of the yellow cube will be larger than the red cube.

What is the Unit of Volume of a Cube?

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

What are the Applications of Volume of a Cube?

Understanding the volume of a cube is useful in a variety of ways. There are numerous items that we use on a daily basis. These include various types of boxes, metal cubes, and so on. All of these are cubes, but they differ in size and texture. The sides of the cubes determine the size of the cubes.

Steel or metal cubes are also used for many industrial purposes.

The understanding of volume is essential in calculating the mass of any object. Suppose if we need to find the mass of a metal cube, we can find the same, if we know the density of the material as: \({\text{mass}} = {\text{volume}} \times {\text{density}}\)

Solved Examples – Volume of a Cube

Q.1. Find the volume of the cube, having the sides of length \(5\,\rm{cm}\).
Ans: Given, the length of the sides of the cube is ‘a’ \(5\,\rm{cm}\)
We know, the volume of a cube \(= {\left(\rm{{side}}\right)^3}\)
Therefore, volume \( = {(5)^3}{\text{c}}{{\text{m}}^3}\)
\(V = 125\,\rm{c{m^3}}\)
Hence, the volume of the cube is \(125\,\rm{c{m^3}}\).

Q.2. Find the length of the sides of the cube if its volume is \(216\,\rm{c{m^3}}\).
Ans: Given, the volume of the cube \(216\,\rm{c{m^3}}\)
Let the length of the sides is \(a\).
We know, The volume of a cube \(= {\left(\rm{{side}}\right)^3}\)
Substituting the value, we get, \({a^3} = 216\)
\(\Rightarrow a = \sqrt[3]{{216}} \Rightarrow a = 6\,{\text{cm}}\)
Thus, the side of the cube is \(6\,\rm{cm}\).

Q.3. Find the diagonal of a cube whose volume is \(343\,\rm{c{m^3}}\)
Ans: We know, \(\frac{{{d^3}}}{{3\sqrt 3 }} = 343\)
\(\Rightarrow {d^3} = 3\sqrt 3 \times 343\)
\(\Rightarrow = \sqrt[3]{{3\sqrt 3 \times 343}} \Rightarrow d = 7\sqrt 3 \,{\text{cm}}\)
Hence, the diagonal is \(7\sqrt 3 \,{\text{cm}}\)

Q.4. If the diagonal is \(4\sqrt 3 \,\rm{cm}\). Find the volume of the cube.
Ans: We know, \(V = \frac{{{d^3}}}{{3\sqrt 3 }} \Rightarrow V = \frac{{{{(4\sqrt 3 )}^3}}}{{3\sqrt 3 }} = \frac{{64 \times 3\sqrt 3 }}{{3\sqrt 3 }} \Rightarrow V = 64\,{\text{c}}{{\text{m}}^3}\)
Hence, the volume of the cylinder is \(64\,{\text{c}}{{\text{m}}^3}\).

Q.5. If the total surface area of a cube \(24\,\rm{c{m^2}}\). Find the volume of the cube.
Ans: Let us assume side of the cube is \(a\)
We know, the total surface area of a cube is \(6{a^2}.\)
\(6{a^2} = 24 \Rightarrow {a^2} = \frac{{24}}{6}\)
\( \Rightarrow {a^2} = 4 \Rightarrow a = 2\,{\text{cm}}\)
Thus, the volume is \({({\text{side}})^3} = {(2)^3} = 8\,{\text{c}}{{\text{m}}^3}\).

Summary

From this article, we have learned that a cube is a three-dimensional object whose all sides are equal, and it has six square shape surfaces and twelve edges, and eight vertices. The three dimensions of the cube are length, breadth, and height and there are different uses of the cube in our real life. Also, the volume of a cube helps us to find the mass if the density is given.

Frequently Asked Questions (FAQs)

Frequently asked questions related to cube and volume of a cube is listed as follows:

Q.1. How do we get the volume formula of a cube?
Ans: If one of the sides of a cube is \(a\;\rm{{unit}}\) then its volume will be \({\left(a \right)^3}\rm{{unit^3}}.\)

Q.2. What is the volume of a \(3 \times 3 \times 3\) cube?
Ans: A \(3 \times 3 \times 3\) cube means, the length of the sides of the cube is \(3\,\rm{units}\) or is \(\rm{length} = \rm{breadth} = \rm{height} = 3\,\rm{units}\) and the volume of it will be \(3 \times 3 \times 3 = 27\,\rm{{unit^3}}\).

Q.3. Is the volume of a cuboid and volume of cube formulae the same?
Ans: We are using the same concept for both cases that is \(\rm{length} \times \rm{breadth} \times \rm{height}\). In a cuboid, all the sides are not the same but in a cube, all the sides are the same.

Q.4. What is the formula for volume?
Ans: The basic formula used to calculate the volume of a three-dimensional geometric shape is given by: \(\rm{area}\,\rm{of}\,\rm{the}\,\rm{base} \times \rm{height}.\)

Q.5. How do you teach volume of a cube?
Ans: We can explain that 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. We can use some real-life examples in the class such as Rubik cubes, dice, etc.

TAKE CLASS 8 MATHS MOCK TEST HERE

Also, Check:

Perimeter of Rectangle	Circumference of Circle
Area of Rectangle	Area of Parallelogram
Area of Rhombus	Area of Square
Area of Right Angle Traingle	Area of Equilateral Traingle

And we hope this article helped you in understanding the concept of volume of cube. However, if you have any questions on how to find the volume of a cube using formula or in general about this article, reach us through the comment box below and we will get back to you as soon as possible.