The sum of the length, breadth and depth of a cuboid is $19 \text{ cm}$ and its diagonal is $5\sqrt{5} \text{ cm}$. It surface area is

### Concept & Formula This problem relies on the algebraic identity connecting the sum of three terms, the sum of their squares, and their product sums. $$ (l+b+h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl) $$ Where $2(lb + bh + hl)$ is the Total Surface Area, and $\sqrt{l^2 + b^2 + h^2}$ is the diagonal. ### Step-by-Step Solution 1. **Identify the given values:** Sum of dimensions: $l + b + h = 19 \text{ cm}$ Diagonal: $\sqrt{l^2 + b^2 + h^2} = 5\sqrt{5} \text{ cm}$ 2. **Square the diagonal to find the sum of squares:** $l^2 + b^2 + h^2 = (5\sqrt{5})^2 = 25 \times 5 = 125$ 3. **Use the algebraic identity:** $(l+b+h)^2 = l^2 + b^2 + h^2 + \text{Total Surface Area}$ $(19)^2 = 125 + \text{Total Surface Area}$ $361 = 125 + \text{Total Surface Area}$ 4. **Solve for the Total Surface Area:** $\text{Total Surface Area} = 361 - 125 = 236 \text{ cm}^2$ ### Exam Strategy & Shortcut Memorize the relationship: $\text{Surface Area} = (\text{Sum of sides})^2 - (\text{Diagonal})^2$. Plug in the numbers instantly: $19^2 - (5\sqrt{5})^2 = 361 - 125 = 236$. ### Common Pitfall Forgetting the square root in the diagonal formula and equating $l^2+b^2+h^2$ directly to $5\sqrt{5}$, or forgetting to square the $5$ when expanding $(5\sqrt{5})^2$. ### Final Answer Therefore, the correct answer is **236 cm$^2$**.