# Volume of a Cube

The volume of a cube is the space it takes up in the three-dimensional plane. It is measured in cubic units such as m3, cm3, mm3, ft3, or in3. The volume of a cube determines how big it is.

## Formulas

### With Edge Length

The basic or common formula to determine the volume of a cube is:

Let us solve some examples involving the above formula.

Find volume of a 5 x 5 cube.

Solution:

As we know,
The edge length determines the size of a cube,
Volume (V) = a3, here a = 5 units
= 53
= 125 cubic units

What is the volume of a 9 inch cube?

Solution:

As we know,
Volume (V) = a3, here a = 9 in
= 93
= 729 in3

Calculate the volume of a 40ft high cube container.

Solution:

As we know,
The length, breadth, and height of a cube is same,
Volume (V) = a3, here a = 40 ft
∴ V = 403
= 64000 ft3

### With Diagonal

The formula to get the volume of a cube when the diagonal is known is:

Derivation

Here we will use the equation of the diagonal to derive the formula of volume of a cube with diagonal.

Length of Space Diagonal (d) = ${\sqrt{3}a}$ — (1)

The equation for volume of a cube with edge length is V = a3, — (2)

Now, replacing the value of a in eqn (1)

a = d/√3, from (1)

V = a3 = (d/√3)3 replacing the value of a from (1)

= d3/(√3 × √3 × √3)

= √3d3/9

∴ V = ${\dfrac{\sqrt{3}}{9}d^{3}}$

(V = √3d3/9)

Calculate the volume of a cube given its diagonal 3 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{\sqrt{3}}{9}d^{3}}$, here d = 3 cm
∴ V = √3 × 33/9
= 5.2 cm3

finding the EDGE of a cube when the VOLUME is known

Calculate the side length of a cube given its volume 343 cubic millimeters.

Solution:

Side length of a cube is actually its edge length. Oftentimes the edge is termed as side length.
So, we will use an alternative formula to find the edge with the volume,
${a=\sqrt[3] {V}}$, here V = 343 mm3
${a=\sqrt[3] {343}}$
= 7 mm