To find the volume of a cube when given its surface area, we can follow these steps:
1. Understand the relationship between surface area and side length:
The surface area [tex]\( A \)[/tex] of a cube can be expressed in terms of its side length [tex]\( a \)[/tex] using the formula:
[tex]\[
A = 6a^2
\][/tex]
Given that the surface area [tex]\( A \)[/tex] is 54 cm², we can set up the equation:
[tex]\[
54 = 6a^2
\][/tex]
2. Solve for the side length [tex]\( a \)[/tex]:
To find the side length, we need to isolate [tex]\( a \)[/tex] in the equation:
[tex]\[
a^2 = \frac{54}{6}
\][/tex]
Simplify the right-hand side:
[tex]\[
a^2 = 9
\][/tex]
Taking the square root of both sides to solve for [tex]\( a \)[/tex]:
[tex]\[
a = \sqrt{9}
\][/tex]
[tex]\[
a = 3 \, \text{cm}
\][/tex]
3. Calculate the volume of the cube:
The volume [tex]\( V \)[/tex] of a cube can be found using the formula:
[tex]\[
V = a^3
\][/tex]
Substituting the side length [tex]\( a = 3 \)[/tex] cm into the formula:
[tex]\[
V = 3^3
\][/tex]
[tex]\[
V = 27 \, \text{cm}^3
\][/tex]
Therefore, the volume of the cube is [tex]\( 27 \)[/tex] cubic centimeters.
