The sum of length, breadth and height of a cuboid is 12 cm long and its diagonalís 8 cm. The total surface area of the cuboid is :

Options

A. 88 sq cm
B. 85 sq cm
C. 90 sq cm
D. 80 sq cm

Answer

To solve the problem step by step, we will use the given information about the cuboid: the sum of its length (L), breadth (B), and height (H) is 12 cm, and its diagonal is 8 cm. We need to find the total surface area of the cuboid.

### Step 1: Set up the equations
We know from the problem that:
1. \( L + B + H = 12 \) (Equation 1)
2. The diagonal of the cuboid can be expressed as:
\[ \sqrt{L^2 + B^2 + H^2} = 8 \]
Squaring both sides gives:
\[ L^2 + B^2 + H^2 = 64 \quad (Equation 2) \]

### Step 2: Use the identity for squares
We can use the identity:
\[ (L + B + H)^2 = L^2 + B^2 + H^2 + 2(LB + BH + HL) \]
Substituting the values from Equation 1 and Equation 2:
\[ 12^2 = 64 + 2(LB + BH + HL) \]
Calculating \(12^2\):
\[ 144 = 64 + 2(LB + BH + HL) \]

### Step 3: Rearranging the equation
Now, rearranging the equation:
\[ 144 - 64 = 2(LB + BH + HL) \]
\[ 80 = 2(LB + BH + HL) \]
Dividing both sides by 2:
\[ LB + BH + HL = 40 \quad (Equation 3) \]

### Step 4: Calculate the total surface area
The total surface area (S) of a cuboid is given by the formula:
\[ S = 2(LB + BH + HL) \]
Substituting the value from Equation 3:
\[ S = 2 \times 40 = 80 \text{ cm}^2 \]

### Final Answer
The total surface area of the cuboid is \( \boxed{80 \text{ cm}^2} \).

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