If the edge of a cube is $12 \, cm$,find its volume (in $cm^3$).

$A$ pyramid is a solid figure,the base of which is

The surface area and volume of a cube with an edge length of $x \ cm$ are numerically equal. Find the value of $x$.

If $V$ is the volume of a cuboid of dimensions $a, b, c$ and $S$ is its total surface area,then prove that $\frac{1}{V} = \frac{2}{S} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)$.

The length,breadth,and height of a hall are $7\, m$,$5\, m$,and $4\, m$ respectively. It has two doors measuring $3\, m \times 2\, m$ each and four windows measuring $2\, m \times 1\, m$ each. Find the cost of whitewashing all the four walls and the ceiling of the hall at the rate of ₹$200$ per $m^2$.

Lateral surface area of a cuboid $= \ldots \ldots \ldots$