Perimeter of a square is 24 cm and length of a rectangle is 8 cm. If the perimeters of the square and the rectangle are equal, then the area (in sq cm) of the rectangle is:

Options

A. 64
B. 16
C. 24
D. 32

Answer

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

### Step 1: Understand the given information
We know that:
- The perimeter of the square is 24 cm.
- The length of the rectangle (l) is 8 cm.
- The perimeters of the square and the rectangle are equal.

### Step 2: Calculate the side length of the square
The formula for the perimeter (P) of a square is given by:
\[ P = 4 \times \text{side} \]

Given that the perimeter of the square is 24 cm, we can set up the equation:
\[ 4 \times \text{side} = 24 \]

To find the side length, we divide both sides by 4:
\[ \text{side} = \frac{24}{4} = 6 \, \text{cm} \]

### Step 3: Set up the equation for the rectangle's perimeter
The formula for the perimeter of a rectangle is given by:
\[ P = 2l + 2b \]
where \( l \) is the length and \( b \) is the breadth.

Since the perimeters of the square and rectangle are equal, we have:
\[ 2l + 2b = 24 \]

### Step 4: Substitute the known length into the perimeter equation
We know the length \( l = 8 \, \text{cm} \). Substituting this into the perimeter equation:
\[ 2(8) + 2b = 24 \]

This simplifies to:
\[ 16 + 2b = 24 \]

### Step 5: Solve for the breadth (b)
Now, we will isolate \( b \) by subtracting 16 from both sides:
\[ 2b = 24 - 16 \]
\[ 2b = 8 \]

Next, divide both sides by 2:
\[ b = \frac{8}{2} = 4 \, \text{cm} \]

### Step 6: Calculate the area of the rectangle
The area (A) of a rectangle is calculated using the formula:
\[ A = l \times b \]

Substituting the values of length and breadth:
\[ A = 8 \times 4 \]
\[ A = 32 \, \text{cm}^2 \]

### Final Answer
The area of the rectangle is \( 32 \, \text{cm}^2 \).

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