The sum of length, breadth and depth of a cuboid is 20 cm and the length of its diagonal is 10 cm, then find the surface area of the cuboid.

Options

A. `150 cm^2`
B. `125 cm^2`
C. `300 cm^2`
D. `145 cm^2`

Answer

To solve the problem step by step, we need to find the surface area of a cuboid given the sum of its length, breadth, and height, as well as the length of its diagonal.

### Step 1: Define the variables
Let:
- \( L \) = length of the cuboid
- \( B \) = breadth of the cuboid
- \( H \) = height (or depth) of the cuboid

From the problem, we know:
1. \( L + B + H = 20 \) cm (Equation 1)
2. The length of the diagonal \( D = 10 \) cm, which gives us \( D^2 = L^2 + B^2 + H^2 \). Therefore, \( L^2 + B^2 + H^2 = 100 \) (Equation 2)

### Step 2: Use the equations to find \( L^2 + B^2 + H^2 \)
We can square Equation 1:
\[ (L + B + H)^2 = 20^2 \]
This expands to:
\[ L^2 + B^2 + H^2 + 2(LB + BH + HL) = 400 \]
Substituting Equation 2 into this:
\[ 100 + 2(LB + BH + HL) = 400 \]
Now, isolate \( 2(LB + BH + HL) \):
\[ 2(LB + BH + HL) = 400 - 100 \]
\[ 2(LB + BH + HL) = 300 \]
Dividing by 2:
\[ LB + BH + HL = 150 \quad (Equation 3) \]

### Step 3: Calculate the surface area
The formula for the surface area \( S \) of a cuboid is given by:
\[ S = 2(LB + BH + HL) \]
Using Equation 3:
\[ S = 2 \times 150 = 300 \text{ cm}^2 \]

### Final Answer
The surface area of the cuboid is \( 300 \text{ cm}^2 \).

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