From each of the four corners of a rectangular sheet of cardboard of dimensions 25 cm × 20 cm, a square of side 2 cm is cut off. The remaining piece is then folded up along the cut edges to form an open rectangular box (without a lid). What is the volume (in cm³) of this box?

Introduction / Context:
This is a standard problem involving converting a flat rectangular sheet into an open box by cutting equal squares from each corner and folding up the sides. The cut-out squares determine the height of the box, and the remaining central rectangle determines the base dimensions. To find the volume of the resulting box, we calculate the length, breadth, and height of the box and then apply the volume formula for a cuboid (rectangular box).

Given Data / Assumptions:

• Original sheet dimensions: length = 25 cm, breadth = 20 cm.
• From each corner, a square of side 2 cm is removed.
• The sheet is folded along the cut edges to form an open box (no lid).
• The height of the box equals the side of the square cut, that is 2 cm.
• We must find the volume of the resulting box in cubic centimetres.

Concept / Approach:
When squares are cut from each corner and the sides are folded up, the length and breadth of the base reduce by twice the side of the square cut, because we remove 2 cm at each end of the dimension. The height of the resulting box is the same as the side of the square. Once we know the effective length, breadth, and height, we use volume = length * breadth * height for the box. The problem therefore reduces to simple subtraction and multiplication.

Step-by-Step Solution:
Original length = 25 cm, original breadth = 20 cm.Side of each cut-out square = 2 cm.When we cut and fold, the height of the box = 2 cm.New length of the box base = 25 − 2 − 2 = 21 cm.New breadth of the box base = 20 − 2 − 2 = 16 cm.Height of the box = 2 cm.Volume of the box = length * breadth * height.So volume = 21 * 16 * 2.First 21 * 16 = 21 * (8 * 2) = (21 * 8) * 2 = 168 * 2 = 336.Now multiply by height 2: 336 * 2 = 672 cm³.

Verification / Alternative check:
We can multiply systematically: 25 − 4 = 21 and 20 − 4 = 16. Then 21 * 16 = (20 + 1) * 16 = 320 + 16 = 336, and times 2 gives 672. The calculations are simple and consistent. Also, since the original rectangle is 25 * 20 = 500 cm², the base 21 * 16 = 336 cm² is smaller due to the removed strips of width 2 cm along each side, which matches the folding logic. This confirms that the dimensions used are realistic and the volume is correctly computed.

Why Other Options Are Wrong:
The volume 828 cm³ would require at least one dimension to be larger than what is available after cutting. The value 500 cm³ would correspond to using the original area as volume with a height of 1 cm, which is incorrect. The value 1000 cm³ is far too large for these dimensions and height. The value 420 cm³ would come from incorrect length or breadth after cutting. Only 672 cm³ matches the exact calculation for the reduced base and given height.

Common Pitfalls:
Common mistakes include subtracting only 2 cm instead of 4 cm from each dimension, forgetting that the cut is taken from both ends. Others may misinterpret the height and think it is still the original thickness or some other value. Some also confuse area with volume and may incorrectly stop at 21 * 16 instead of multiplying by the height. Remembering that all three dimensions are needed for volume is crucial.

Final Answer:
The volume of the open box formed from the cardboard sheet is 672 cm³.