The sum of the length, breadth and height of a cuboid is 38 cm and the length of its diagonal is 22 cm. Find the surface area of the cuboid.

Answer

To find the surface area of the cuboid given the sum of its dimensions and the length of its diagonal, we can follow these steps:

### Step 1: Define the variables
Let:
- Length = \( l \)
- Breadth = \( b \)
- Height = \( h \)

### Step 2: Set up the equations
From the problem statement, we have two equations:
1. The sum of the length, breadth, and height:
\[ l + b + h = 38 \quad \text{(Equation 1)} \]
2. The length of the diagonal:
\[ \sqrt{l^2 + b^2 + h^2} = 22 \]
Squaring both sides gives:
\[ l^2 + b^2 + h^2 = 22^2 = 484 \quad \text{(Equation 2)} \]

### Step 3: Expand the square of the sum of dimensions
We can use the identity:
\[ (l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl) \]
Substituting Equation 1 into this identity:
\[ 38^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl) \]
Calculating \( 38^2 \):
\[ 1444 = l^2 + b^2 + h^2 + 2(lb + bh + hl) \]
Now substitute Equation 2 into this equation:
\[ 1444 = 484 + 2(lb + bh + hl) \]

### Step 4: Solve for \( lb + bh + hl \)
Rearranging the equation gives:
\[ 1444 - 484 = 2(lb + bh + hl) \]
\[ 960 = 2(lb + bh + hl) \]
Dividing both sides by 2:
\[ lb + bh + hl = 480 \quad \text{(Equation 3)} \]

### Step 5: Calculate the surface area
The surface area \( S \) of a cuboid is given by:
\[ S = 2(lb + bh + hl) \]
Substituting Equation 3 into this formula:
\[ S = 2 \times 480 = 960 \, \text{cm}^2 \]

### Final Answer
The surface area of the cuboid is \( 960 \, \text{cm}^2 \).

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