Cuboid from sum of edges and space diagonal — find surface area: The sum of the length, breadth, and depth (height) of a cuboid is 19 cm, and its space diagonal is 5√5 cm. Find the total surface area of the cuboid.

Introduction / Context:
This problem asks for the surface area of a cuboid when the sum of its three edges (l + b + h) and its space diagonal are known. It relies on two identities that connect edges, diagonal, and pairwise products.

Given Data / Assumptions:

• Let l, b, h be the edges in centimeters.
• l + b + h = 19 cm.
• Space diagonal d = 5√5 cm ⇒ d^2 = 125.
• Total surface area S = 2(lb + bh + hl).

Concept / Approach:
We use the identities: (l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl), and d^2 = l^2 + b^2 + h^2. Eliminating l^2 + b^2 + h^2 allows us to solve directly for (lb + bh + hl) and hence S.

Step-by-Step Solution:
(l + b + h)^2 = 19^2 = 361d^2 = l^2 + b^2 + h^2 = 125361 = 125 + 2(lb + bh + hl) ⇒ lb + bh + hl = (361 − 125)/2 = 118Surface area S = 2(lb + bh + hl) = 2 * 118 = 236 cm2

Verification / Alternative check:
No dimensions are individually needed; identities suffice and are standard for cuboids.

Why Other Options Are Wrong:
361 cm2 confuses (l + b + h)^2 with area; 125 cm2 uses d^2 incorrectly; 486 cm2 doubles a wrong intermediate.

Common Pitfalls:
Forgetting to multiply the pairwise sum by 2 at the end; mixing up d^2 with (l + b + h)^2.

Final Answer:
236 cm2