Question

Find the lateral surface of the cube if its diagonal is \[\sqrt{6}\]cm.

Answer 1

Hint: First assume the edge of the cube as a cm then calculate the edge of the cube as we know the formula for diagonal of a cube with edge a cm. After getting the edge of the cube substitute the value of edge of the cube obtained in the formula of lateral surface area of the cube.

Complete step-by-step answer:
Given that the diagonal of cube is \[\sqrt{6}\]cm
Let the edge of cube be a cm
We know that the diagonal of a cube with side of length a cm is given by \[\sqrt{3}a\]
\[\sqrt{3}a=\sqrt{6}\]. . . . . . . . . . . . . . . . . . . (1)
\[\Rightarrow a=\sqrt{2}\]. . . . . . . . . . . . . . . . . . . (2)
So the length of edge of the cube is \[a=\sqrt{2}\]cm.
We know that the lateral surface of the cube is given by \[4{{a}^{2}}\]
By substituting the a value we will get
\[=4{{\left( \sqrt{2} \right)}^{2}}\]. . . . . . . . . . . . . . . . . . . (3)
\[=4\times 2\]
\[=8c{{m}^{2}}\]
So, the obtained value of lateral surface of cube \[=8c{{m}^{2}}\]

Note: The lateral surface area of solid is the sum of area of the solid without bases so lateral surface area is given by the formula \[4{{a}^{2}}\] and total surface area of solid is sum of areas of all faces and total surface area of cube is given by \[6{{a}^{2}}\]. The cube is a three dimensional object bounded by six square faces.