A circle and a rectangle have the same perimeter.The sides of the rectangle are 18 cm and 26 cm. What is the area of the circle ?

Options

A. `88 cm^2`
B. `154 cm^2`
C. `616 cm^2`
D. `1250 cm^2`

Answer

To find the area of the circle that has the same perimeter as a rectangle with sides 18 cm and 26 cm, we can follow these steps:

### Step 1: Calculate the perimeter of the rectangle.
The formula for the perimeter (P) of a rectangle is:
\[ P = 2 \times (L + B) \]
where L is the length and B is the breadth.

Given:
- Length (L) = 26 cm
- Breadth (B) = 18 cm

Calculating the perimeter:
\[ P = 2 \times (26 + 18) \]
\[ P = 2 \times 44 \]
\[ P = 88 \text{ cm} \]

### Step 2: Set the perimeter of the circle equal to the perimeter of the rectangle.
The perimeter (circumference) of a circle is given by:
\[ C = 2 \pi r \]
where r is the radius of the circle.

Since the perimeter of the rectangle is equal to the perimeter of the circle:
\[ 2 \pi r = 88 \]

### Step 3: Solve for the radius (r).
Rearranging the equation to find r:
\[ r = \frac{88}{2 \pi} \]
\[ r = \frac{88}{2 \times \frac{22}{7}} \]
\[ r = \frac{88 \times 7}{44} \]
\[ r = 14 \text{ cm} \]

### Step 4: Calculate the area of the circle.
The area (A) of a circle is given by:
\[ A = \pi r^2 \]

Substituting the value of r:
\[ A = \pi \times (14)^2 \]
\[ A = \pi \times 196 \]
Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{22}{7} \times 196 \]

### Step 5: Simplify the area calculation.
Calculating:
\[ A = \frac{22 \times 196}{7} \]
\[ A = \frac{4312}{7} \]
\[ A = 616 \text{ cm}^2 \]

### Final Answer:
The area of the circle is \( 616 \text{ cm}^2 \).

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