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:
Volume of a Cube
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:
Volume of a Cube with Diagonal
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
