A rectangular cardboard sheet has four equal squares removed from its corners as shown in the figure. The obtained piece is then used to make a box. The base of the box has a length of 15 cm and a width of 12 cm. The volume of the box is 540 cm³.

What was the original area of the sheet of cardboard?

Suppose the height of the box is h cm.
Length of the box = 15 cm
Breadth of the box = 12 cm
So volume of the box = $15\times 12\times h = 540 {\mathrm{cm}}^3$
$$\begin{gathered} \Rightarrow 180h = 540 \\ \Rightarrow h = \frac{540}{180} = 3 \end{gathered}$$
So height of the box = 3 cm
Now this height of the box is the length of each corner of the rectangular cardboard sheet.
So original length of the rectangular cardboard = 15 cm + 3 cm + 3 cm = 21 cm
And breadth of the rectangular sheet = 12 cm + 3 cm + 3 cm = 18 cm
So original area of rectangular cardboard = $21 \mathrm{cm} \times 18 \mathrm{cm} = 378 {\mathrm{cm}}^2$
Therefore the original area of the sheet of cardboard is 378 sq. cm.