Volume of a Cube – Derivation, Formula & Examples

Every 3D object occupies some space. The space occupied by a 3D object is called its volume. Volume is also referred to as the capacity of an object. It is measured in terms of cubic units, such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$, etc. 

The concept of the volume of an object helps us to determine the amount required to fill that object, e.g., the amount of water needed to fill a bottle or the amount of milk that a container can hold, etc. 

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

Cube – A 3D Solid Shape

A cube(or regular hexahedron) is a 3D solid shape with six square faces and all the edges (or sides) of a cube are of the same length. The shape consists of six square faces, eight vertices, and twelve edges. The length, width(or breadth), and height are of the same measurement in a cube since the 3D figure is a square that has all sides of the same length.

A cube has the following properties:

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

Difference Between Cube and Cuboid

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

What is the Volume of a Cube?

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

You can find the volume of a cube in either of the following ways.

• by using the length of the side of a cube
• by using the length of the diagonal of a cube

Volume of Cube Formula – Volume of a Cube Using the Side of a Cube

If the length of each edge(side) of a cube is $a$ unit, then the formula for calculating its volume is $a \times a \times a = a^{3} unit^{3}$.

Note: Volume of a cuboid of length, width, and height as $l$, $w$ and $h$ respectively is $lwh$. In the case of a cube, $l = w = h = a$, therefore, the volume of a cube is $a \times a \times a = a^{3}$.

Volume of Cube Formula – Volume of a Cube Using the Diagonal of a Cube

If the length of the diagonal of a cube is $d$ unit, then the formula for calculating its volume is $\frac {\sqrt{3}d^{3}}{9}$.

Let’s consider a cube of the length of edge $a$ unit and the length of diagonal as $d$ unit, then $d = \sqrt {a^{2} + a^{2} + a^{2}} = \sqrt{3a^{2}} => d = \sqrt{3} a => a = \frac {d}{\sqrt{3}}$

Therefore, the formula for the volume of a cube becomes $\left(\frac {d}{\sqrt{3}} \right)^{3} = \frac {d^{3}}{3 \sqrt{3}} = \frac {d^{3}}{3 \sqrt{3}} \times \frac {\sqrt {3}}{\sqrt {3}} = \frac {\sqrt{3} d^{3}}{9}$.

Finding the Length of Diagonal of a Cube

Consider a cube $ABCDEFGH$ of edge length $a$.

One of the diagonals of a cube is $AG$.

Consider a right $\triangle EFG$, right-angled at $F$.

Using Pythagoras theorem, we get $EG^{2} = EF^{2} + FG^{2} => EG^{2} = a^{2} + a^{2} => EG^{2} = 2a^{2}$.

Now, consider right $\triangle AEG$, right-angled at $E$.

Again using Pythagoras theorem, we get $AG^{2} = AE^{2} + EG^{2} => AG^{2} = a^{2} + 2a^{2} => AG^{2} = 3a^{2} => AG = \sqrt{3}a$.

Therefore, $\text{Diagonal} = \sqrt{3} \times \text{Side}$.

Examples

Ex 1: Find the volume of a cube whose side length is $4 cm$.

Length of side of a cube $a = 4 cm$.

Volume of a cube = $a^{3} = 4^{3} = 64 cm^{3}$.

Ex 2: Find the volume of a cube whose diagonal is $9 in$.

Length of diagonal of a cube $d = 9 in$.

Volume of a cube = $ \frac {\sqrt{3} d^{3}}{9} = \frac {\sqrt{3} \times 9^{3}}{9} = 9^{2} \sqrt{3} = 81\sqrt{3} in^{3}$.

Ex 3: A cube with an edge of $7 cm$ and a cuboid measuring $7 cm \times 4 cm \times 8 cm$ are kept on a table. Which shape has more volume?

Length of edge of a cube $a = 7 cm$

Volume of a cube = $a^{3} = 7^{3} = 343 cm^{3}$

Dimensions of a cuboid are $l = 7 cm$, $w = 4 cm$ and $h = 8 cm$

Volume of a cuboid = $lbh = 7 \times 4 \times 8 = 224 cm^{3}$.

Ex 4: A box measures $11 cm \times 10 cm \times 8 cm$. How many dice can fit in this box if the dice are cubes with sides of length $2 cm$?

Dimensions of cuboidal box are $l = 11 cm$, $w = 10 cm$, $h = 8 cm$

Volume of cuboidal box = $lwh = 11 \times 10 \times 8 = 880 cm^{3}$

Length of an edge of a cubical dice $a = 2 cm$

Volume of one cubical dice = $a^{3} = 2^{3} = 8 cm^{3}$

Number of dice that can fit in a box = $\frac {880}{8} = 110$.

Ex 5: If the volume of a cube is $3375 m^{3}$, what is the length of each side?

Volume of a cube = $3375 m^{3}$

Let the length of an edge of a cube = $a$

$a^{3} = 3375 => a = \sqrt[3]{3375} => a = 15 m$

Therefore, the length of each side of a cube is $15 m$.

Note: $3375 = 3 \times 3 \times 3 \times 5 \times 5 \times 5 = 3^{3} \times 5^{3} = \left(3 \times 5 \right)^{3} = 15^{3}$. Therefore, $\sqrt[3]{3375} = 15$.

Ex 6: If the edge of a cube is doubled, then how much does the volume increase?

Let the length of an edge of a cube be $a$

Volume of cube = $a^{3}$.

Length of an edge of a cube after increase = $2a$

Volume of cube = $\left(2a \right)^{3} = 2^{3} \times a^{3} = 8a^{3}$.

Therefore, the volume of a cube increases eight times when the edge of the cube is doubled.

Practice Problems

1. What is the volume of a cube whose sides are $12$ cm each?
2. The volume of a cube is $343 m^{3}$. Find the side lengths of the cube.
3. The edge of a Rubik’s cube is $0.04 m$. Find the volume of the Rubik’s cube.
4. Cubical bricks of length $6 cm$ are stacked such that the height, width, and length of the stack is $30 cm$ each. Find the number of bricks in the stack.
5. How many cubical boxes of edge length $4 cm$ can be packed in a large cubical case of length $8 cm$?

FAQs

Conclusion

The volume of a cube is the number of cubic units, occupied by the cube completely and is calculated by cubing the length of an edge of a cube, i.e., for a cube of edge length $a$, its volume is $a^{3}$. One can also calculate the volume of a cube if its diagonal length is known using the formula $\frac {\sqrt{3} d^{3}}{9}$.

Recommended Reading

• Surface Area of a Cylinder(Definition, Formulas & Examples)
• Surface Area of a Cone(Definition, Formulas & Examples)
• Surface Area of A Cube (Definition, Formula & Examples)
• Surface Area of Cuboid (Definition, Formula & Examples)
• Area of Rectangle – Definition, Formula & Examples
• Area of Square – Definition, Formula & Examples
• Area of a Triangle – Formulas, Methods & Examples
• Area of a Circle – Formula, Derivation & Examples
• Area of Rhombus – Formulas, Methods & Examples
• Area of A Kite – Formulas, Methods & Examples
• Perimeter of a Polygon(With Formula & Examples)
• Perimeter of Trapezium – Definition, Formula & Examples
• Perimeter of Kite – Definition, Formula & Examples
• Perimeter of Rhombus – Definition, Formula & Examples
• Circumference (Perimeter) of a Circle – Definition, Formula & Examples
• Perimeter of Square – Definition, Formula & Examples
• Perimeter of Rectangle – Definition, Formula & Examples
• Perimeter of Triangle – Definition, Formula & Examples
• What Are 2D Shapes – Names, Definitions & Properties

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