Question

The volume of a cube is $343c{{m}^{3}}$. Find its surface area.

Answer 1

Hint: We are given the volume of a cube. We will first find a side of the cube with the help of the formula $V={{a}^{3}}$ and then after finding the side “a”, we will find the surface area of the cube with the formula $TSA=6{{a}^{2}}$.

Complete step by step answer:
Let us first recall the definition and properties of a cube.
A cube is a three dimensional object formed by joining six square faces where three faces meet at each vertex. It has 6 faces, 12 edges, and 8 vertices.
Since all the faces of a cube are squares of the same edge length, the angles formed by the faces with each other are also the same.
The angle formed by the faces of the cube with each other is $\dfrac{\pi }{2}$.

For a cube with edge length “a”,
The volume V is given by $V={{a}^{3}}$
The lateral surface area LSA is given by $LSA=4{{a}^{2}}$
The total surface area TSA is $TSA=6{{a}^{2}}$
The diagonal of the cube is given by $d=\sqrt{3}a$ whereas the diagonal of the side of the cube is given by $\sqrt{2}a$.
Now, according to the question,
Volume is given by $343c{{m}^{3}}$
Then $V=343$
$\begin{align}
  & \Rightarrow {{a}^{3}}=343 \\
 & \Rightarrow {{a}^{3}}={{7}^{3}} \\
 & \Rightarrow a=7cm \\
\end{align}$
Now, the formula for surface area is given by
$TSA=6{{a}^{2}}$
Putting the value of a, we get
$\begin{align}
  & TSA=6{{a}^{2}} \\
 & =6{{\left( 7 \right)}^{2}} \\
 & =294c{{m}^{2}}
\end{align}$

Thus, the surface area is given by $294c{{m}^{2}}$.

Note: We know two types of surface area: lateral surface area and total surface area. The question asks us to find the surface area of the cube but doesn’t mention which one. So, we will always consider total surface area unless and until mentioned otherwise.