Question

The lateral surface area of a cube is ${\text{256}}{{\text{m}}^{\text{2}}}$. The volume of the cube is
A. ${\text{512}}{{\text{m}}^{\text{3}}}$
B. ${\text{64}}{{\text{m}}^{\text{3}}}$
C. ${\text{216}}{{\text{m}}^{\text{3}}}$
D. ${\text{256}}{{\text{m}}^{\text{3}}}$

Answer 1

Hint: We have the lateral surface area of a cube. We can find the length of the side from the lateral surface area using the equation $LSA = 4{a^2}$. Now we will get the length of the side a. Then using the side length, we can find the volume of the cube using the equation ${\text{V = }}{{\text{a}}^{\text{3}}}$ .

Complete step by step answer:

We know that a cube has 6 square surfaces. The lateral surface area of a cube is given by the sum of the 4 lateral square surfaces.
Let a be the side length of the cube then area of each square surface is ${{\text{a}}^{\text{2}}}$. Then lateral surface area is given by ${\text{4}}{{\text{a}}^{\text{2}}}$. But we are given that lateral surface area of the cube is ${\text{256}}{{\text{m}}^{\text{2}}}$.so we get,
$
  {\text{4}}{{\text{a}}^{\text{2}}}{\text{ = 256}} \\
   \Rightarrow {{\text{a}}^{\text{2}}}{\text{ = }}\dfrac{{{\text{256}}}}{{\text{4}}}{\text{ = 64}} \\
 $
Taking the square root, we get,
${\text{a = }}\sqrt {{\text{64}}} {\text{ = 8m}}$
Now we can find the volume of the cube using the equation ${\text{V = }}{{\text{a}}^{\text{3}}}$, so we get,
${\text{V = }}{{\text{8}}^{\text{3}}}{\text{ = 512}}{{\text{m}}^{\text{3}}}$
So, the volume of the given cube is ${\text{512}}{{\text{m}}^{\text{3}}}$
Therefore, the correct is answer option A.

Note: A cube has 6 square surfaces. A cube with side length a is given in the figure. Its top and bottom surfaces are A and B respectively. As all the sides of a cube are equal, the surfaces will be identical. The total surface area is the total area of all the surfaces of the cube. From the figure, we have 6 surfaces for a cube and each surface has an area of ${{\text{a}}^{\text{2}}}$. So, the total surface area is given by six times the area of one surface. i.e.,${\text{6}}{{\text{a}}^{\text{2}}}$ Lateral surface area is the surface area excluding the top and bottom surface area. It can be found by removing the area of the top and bottom surface areas from the total surface area. It is given by $LSA = TSA$ - (area of surface A + area of surface B) $ \Rightarrow LSA = 6{a^2} - 2{a^2} = 4{a^2}$. We must use proper units and must do necessary unit conversions if needed.