The sum of length, breadth and heigth of a cuboid is 30 cm and the length of its diagonal is 22 cm. Find the surface area of the cuboid.

Answer

To solve the problem, we need to find the surface area of a cuboid given the sum of its dimensions and the length of its diagonal.

### Step-by-Step Solution:

1. **Let the dimensions be defined:**
Let the length, breadth, and height of the cuboid be denoted as \( L \), \( B \), and \( H \) respectively.

2. **Use the given information:**
We know that:
\[ L + B + H = 30 \quad \text{(1)} \]
and the length of the diagonal is given by:
\[ \sqrt{L^2 + B^2 + H^2} = 22 \quad \text{(2)} \]
Squaring both sides of equation (2):
\[ L^2 + B^2 + H^2 = 22^2 = 484 \quad \text{(3)} \]

3. **Expand the square of the sum of dimensions:**
From equation (1), squaring both sides gives:
\[ (L + B + H)^2 = 30^2 = 900 \]
Expanding this, we have:
\[ L^2 + B^2 + H^2 + 2(LB + BH + HL) = 900 \quad \text{(4)} \]

4. **Substitute equation (3) into equation (4):**
We already know from equation (3) that \( L^2 + B^2 + H^2 = 484 \). Substituting this into equation (4):
\[ 484 + 2(LB + BH + HL) = 900 \]

5. **Solve for \( 2(LB + BH + HL) \):**
Rearranging gives:
\[ 2(LB + BH + HL) = 900 - 484 = 416 \]
Dividing by 2:
\[ LB + BH + HL = 208 \quad \text{(5)} \]

6. **Calculate the surface area of the cuboid:**
The surface area \( S \) of a cuboid is given by the formula:
\[ S = 2(LB + BH + HL) \]
Substituting the value from equation (5):
\[ S = 2 \times 208 = 416 \, \text{cm}^2 \]

### Final Answer:
The surface area of the cuboid is \( 416 \, \text{cm}^2 \).