Question

Diagonal of a cube is 12 cm then its L.S.A is
A) 144 sq.cm
B) 64 sq.cm
C) 192 sq.cm
D) 1728 sq.cm

Answer 1

Hint:
Here, we will use the diagonal of a cube formula to find the length of the side of the cube. We will then use the lateral surface area of the cube to find the L.S.A. The lateral surface area of the cube is the area of four sides.

Formula Used:
We will use the following formula:
1) Diagonal of a cube is given by the formula \[d = \sqrt 3 a\] where \[a\] is the side of the cube.
2) Lateral Surface area of the cube is given by the formula \[L.S.A. = 4{\left( a \right)^2}\] where \[a\] is the side of the cube.

Complete step by step solution:
We are given that the diagonal of a cube is 12 cm.
Diagonal of a cube \[ = 12cm\]
By using the diagonal of a cube formula \[d = \sqrt 3 a\], we get
\[ \Rightarrow \sqrt 3 a = 12\]
Dividing both side by \[\sqrt 3 \], we get
\[ \Rightarrow a = \dfrac{{12}}{{\sqrt 3 }}cm\]
Thus, the side of the cube is \[\dfrac{{12}}{{\sqrt 3 }}cm\].
Now, we will find the L.S.A. of the cube using the L.S.A. formula.
By substituting \[a = \dfrac{{12}}{{\sqrt 3 }}\] of the cube \[L.S.A. = 4{\left( a \right)^2}\], we get
\[ \Rightarrow L.S.A = 4{\left( {\dfrac{{12}}{{\sqrt 3 }}} \right)^2}\]
By rewriting the equation, we get
\[ \Rightarrow L.S.A = 4\left( {\dfrac{{12}}{{\sqrt 3 }} \times \dfrac{{12}}{{\sqrt 3 }}} \right)\]
\[ \Rightarrow L.S.A = 4\left( {\dfrac{{12 \times 12}}{3}} \right)\]
 By multiplying the terms, we get
\[ \Rightarrow L.S.A = 192\] sq.cm
Therefore, the L.S.A. of a cube is \[192\] sq.cm.

Thus, option (C) is the correct answer.

Note:
We know that a cube is a three dimensional figure and it is bounded by squares on all the sides. The properties of the cube include that the Cube has all its faces in a square shape. Thus six faces are in square shape and all the faces or sides have equal dimensions. Each of the vertices meets the three faces and three edges and the edges opposite to each other are parallel. The difference between the square and the cube is square is a two-dimensional figure with two dimensions length and breadth whereas the cube is a three-dimensional figure with three dimensions length, breadth and height.