Question

tatal surface area of a cube is 486cm find the lateral surface area and the diagonal of the cube​

Answer 1

Cube and its Area

A cub is a {eq}3 {/eq} -dimensional object that may be solid or hollow has {eq}12 {/eq} identical sides which are connected to each other at the right angle. Therefore, it has {eq}12 {/eq}-sides ,{eq}8 {/eq}-corners( vertex) and {eq}6 {/eq} square faces. A cube has {eq}4 {/eq} identical diagonals that bisect each other at the center.

Lateral surface Area:

The area of all square faces of the cube except the top and bottom face.

$$\displaystyle S_{l} = 4a^{2} $$

Here {eq}a {/eq} is the side of the cube.

And

Total surface area

$$\displaystyle S_{t} = 6a^{2} $$

Answer and Explanation:

Given:

The total surface area of the cube is {eq}\displaystyle 486~\rm cm^{2} {/eq}

So

$$\begin{align} S_{t} & = 6a^{2} \\[0.2 cm] 6a^{2} & = 486 \\[0.2 cm] a^{2} & = 81 \\[0.2 cm] a & = 9 ~\rm cm \\[0.2 cm] \end{align} $$

Now

The lateral surface area of the cube:

$$\begin{align} S_{l} & = 4a^{2} \\[0.2 cm] & = 4 \times 9^{2} \\[0.2 cm] & = 4 \times 81 \\[0.2 cm] & = 324 ~\rm cm^{2} \\[0.2 cm] \end{align} $$