The sum of length, breadth and height of a cuboid is 22 cm and the length of its body diagonal is 14 cm. What is the surface area of the cuboid?
Solution
✅ Correct Option: 1
Let the sides be $l$, $b$ and $h$. $l+b+h=22$ Using the formula for the body diagonal of a cuboid, $\sqrt{l^2+b^2+h^2}$: $\sqrt{l^2+b^2+h^2}=14$ $\Rightarrow l^2+b^2+h^2=196$ Using the identity $(l+b+h)^2=l^2+b^2+h^2+2(lb+bh+hl)$: $22^2=196+2(lb+bh+hl)$ $\Rightarrow 484-196=2(lb+bh+hl)$ $\Rightarrow 2(lb+bh+hl)=288$ The surface area of a cuboid is exactly $2(lb+bh+hl)$, so the answer is already in front of us and we never need $l$, $b$ and $h$ separately. $\text{Surface area}=288\text{ cm}^2$