Sum of the length, width and depth of a cuboid is s and its diagonal is d. Its total surface area is :

Question

Accepted Answer

To determine the total surface area of a cuboid given that the sum of its length, width, and depth is $ s $ and its diagonal is $ d $, we need to analyze the relationship between these parameters.

1. **Explanation for Option 4: $(s^2 - d^2)$ sq. unit**

The total surface area $ A $ of a cuboid with dimensions $ l $, $ w $, and $ h $ is given by:
   $$
   A = 2(lw + lh + wh)
   $$
   We know:
   $$
   l + w + h = s
   $$
   and the diagonal $ d $ is given by:
   $$
   d = \sqrt{l^2 + w^2 + h^2}
   $$
   Squaring both sides of the diagonal equation, we have:
   $$
   d^2 = l^2 + w^2 + h^2
   $$
   From the identity $(l + w + h)^2 = l^2 + w^2 + h^2 + 2(lw + lh + wh)$, we can substitute:
   $$
   s^2 = l^2 + w^2 + h^2 + 2(lw + lh + wh)
   $$
   Substituting $ d^2 = l^2 + w^2 + h^2 $ into the equation, we get:
   $$
   s^2 = d^2 + 2(lw + lh + wh)
   $$
   Rearranging gives:
   $$
   2(lw + lh + wh) = s^2 - d^2
   $$
   Therefore, the total surface area $ A $ is:
   $$
   A = 2(lw + lh + wh) = s^2 - d^2
   $$
   Thus, Option 4 is correct.

2. **Explanation for why other options are incorrect:**

- **Option 1: $s^2$ sq. unit**

This option suggests that the total surface area is simply the square of the sum of the dimensions. However, this does not account for the relationship between the dimensions and the diagonal. The correct expression involves both $ s^2 $ and $ d^2 $.

- **Option 2: $(s^2 + d^2)$ sq. unit**

This option incorrectly adds $ d^2 $ to $ s^2 $. As derived above, the correct relationship involves subtracting $ d^2 $ from $ s^2 $ to account for the squared dimensions in the diagonal formula.

- **Option 3: $d^2$ sq. unit**

This option suggests that the total surface area is simply the square of the diagonal, which is incorrect. The diagonal only represents the spatial diagonal of the cuboid and does not directly relate to the surface area without considering the sum of the dimensions.

In conclusion, the correct answer is Option 4: $(s^2 - d^2)$ sq. unit, as it accurately represents the total surface area of the cuboid based on the given conditions.