The length of a hall is double its breadth , its height is 3 m . The area of its four walls. (including doores and windows) is `108 m^(2) ` . Find its volume.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Problem
We know that the length (L) of the hall is double its breadth (B), and the height (H) is given as 3 m. The area of the four walls (lateral surface area) is given as 108 m².

### Step 2: Set Up the Relationships
From the problem, we can establish the following relationships:
- \( L = 2B \) (Length is double the breadth)
- \( H = 3 \) m (Height is given)

### Step 3: Use the Formula for Lateral Surface Area
The formula for the lateral surface area (A) of a cuboid is:
\[ A = 2H(L + B) \]
We know that the area of the four walls is 108 m², so we can set up the equation:
\[ 2H(L + B) = 108 \]

### Step 4: Substitute Known Values
Substituting the known values into the equation:
\[ 2 \times 3 \times (L + B) = 108 \]
This simplifies to:
\[ 6(L + B) = 108 \]

### Step 5: Solve for \( L + B \)
Now, divide both sides by 6:
\[ L + B = \frac{108}{6} \]
\[ L + B = 18 \]

### Step 6: Substitute \( L \) in Terms of \( B \)
Since \( L = 2B \), we can substitute this into the equation:
\[ 2B + B = 18 \]
This simplifies to:
\[ 3B = 18 \]

### Step 7: Solve for \( B \)
Now, divide both sides by 3:
\[ B = \frac{18}{3} \]
\[ B = 6 \] m

### Step 8: Find \( L \)
Now that we have \( B \), we can find \( L \):
\[ L = 2B = 2 \times 6 = 12 \] m

### Step 9: Find the Volume
The volume (V) of the cuboid is given by the formula:
\[ V = L \times B \times H \]
Substituting the values we found:
\[ V = 12 \times 6 \times 3 \]

### Step 10: Calculate the Volume
Now, calculate the volume:
\[ V = 72 \times 3 = 216 \] m³

### Final Answer
The volume of the hall is **216 m³**.

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