Area of a rectangle and the area of a circle are equal. If the dimensions of the rectangle are 14cm x 11 cm, then radius of the circle is

Options

A. 21 cm
B. 10.5 cm
C. 14 cm
D. 7 cm.

Answer

To find the radius of the circle when the area of the rectangle and the area of the circle are equal, follow these steps:

### Step 1: Calculate the area of the rectangle
The area of a rectangle is given by the formula:
\[ \text{Area of Rectangle} = \text{length} \times \text{width} \]
Given the dimensions of the rectangle are 14 cm and 11 cm, we can calculate:
\[ \text{Area of Rectangle} = 14 \, \text{cm} \times 11 \, \text{cm} = 154 \, \text{cm}^2 \]

### Step 2: Set the area of the rectangle equal to the area of the circle
According to the problem, the area of the rectangle is equal to the area of the circle. The area of a circle is given by the formula:
\[ \text{Area of Circle} = \pi r^2 \]
Thus, we can set up the equation:
\[ 154 \, \text{cm}^2 = \pi r^2 \]

### Step 3: Substitute the value of \(\pi\)
For simplicity, we can use the approximation \(\pi \approx \frac{22}{7}\). Substituting this into the equation gives:
\[ 154 = \frac{22}{7} r^2 \]

### Step 4: Solve for \(r^2\)
To eliminate the fraction, multiply both sides by 7:
\[ 154 \times 7 = 22 r^2 \]
Calculating the left side:
\[ 1078 = 22 r^2 \]

### Step 5: Isolate \(r^2\)
Now, divide both sides by 22:
\[ r^2 = \frac{1078}{22} \]
Calculating the right side:
\[ r^2 = 49 \]

### Step 6: Find \(r\)
To find \(r\), take the square root of both sides:
\[ r = \sqrt{49} = 7 \, \text{cm} \]

### Conclusion
The radius of the circle is \(7 \, \text{cm}\).

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