If circumference of first circle is 132 cm and circumference of second circle is 110 cm then find the difference between area of both the circle?

Options

A. 423.5 `cm^(2)`
B. 412.5 `cm^(2)`
C. 420 `cm^(2)`
D. 422.4 `cm^(2)`

Answer

To find the difference between the areas of two circles given their circumferences, we can follow these steps:

### Step 1: Find the radius of the first circle
The circumference \( C \) of a circle is given by the formula:
\[ C = 2 \pi r \]
For the first circle, the circumference is 132 cm. Therefore, we can set up the equation:
\[ 2 \pi R_1 = 132 \]
Substituting \( \pi \) with \( \frac{22}{7} \):
\[ 2 \times \frac{22}{7} \times R_1 = 132 \]
Now, we can solve for \( R_1 \):
\[ \frac{44}{7} R_1 = 132 \]
Multiplying both sides by \( \frac{7}{44} \):
\[ R_1 = 132 \times \frac{7}{44} = 21 \text{ cm} \]

### Step 2: Find the radius of the second circle
Similarly, for the second circle with a circumference of 110 cm:
\[ 2 \pi R_2 = 110 \]
Substituting \( \pi \) with \( \frac{22}{7} \):
\[ 2 \times \frac{22}{7} \times R_2 = 110 \]
Solving for \( R_2 \):
\[ \frac{44}{7} R_2 = 110 \]
Multiplying both sides by \( \frac{7}{44} \):
\[ R_2 = 110 \times \frac{7}{44} = 17.5 \text{ cm} \]

### Step 3: Calculate the area of the first circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
For the first circle:
\[ A_1 = \pi R_1^2 = \frac{22}{7} \times (21)^2 \]
Calculating \( (21)^2 = 441 \):
\[ A_1 = \frac{22}{7} \times 441 = \frac{9702}{7} = 1386 \text{ cm}^2 \]

### Step 4: Calculate the area of the second circle
For the second circle:
\[ A_2 = \pi R_2^2 = \frac{22}{7} \times (17.5)^2 \]
Calculating \( (17.5)^2 = 306.25 \):
\[ A_2 = \frac{22}{7} \times 306.25 = \frac{6747.5}{7} = 962.5 \text{ cm}^2 \]

### Step 5: Find the difference between the areas
Now, we can find the difference between the areas of the two circles:
\[ \text{Difference} = A_1 - A_2 = 1386 - 962.5 = 423.5 \text{ cm}^2 \]

### Final Answer
The difference between the areas of both circles is \( 423.5 \text{ cm}^2 \).

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