If the circumference of a circle is increased by 20% then the area will be increased by

Options

A. 44
B. 50
C. 60
D. 42

Answer

To solve the problem of how much the area of a circle increases when its circumference is increased by 20%, we can follow these steps:

### Step 1: Understand the relationship between circumference and radius.
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

### Step 2: Calculate the new circumference after a 20% increase.
If the circumference is increased by 20%, the new circumference \( C' \) can be calculated as:
\[ C' = C + 0.2C = 1.2C \]
Substituting the formula for circumference:
\[ C' = 1.2 \times 2\pi r = \frac{12}{10} \times 2\pi r = \frac{12\pi r}{5} \]

### Step 3: Relate the new circumference to the new radius.
Let the new radius be \( R \). The new circumference can also be expressed as:
\[ C' = 2\pi R \]
Setting the two expressions for \( C' \) equal gives:
\[ 2\pi R = \frac{12\pi r}{5} \]
Dividing both sides by \( 2\pi \):
\[ R = \frac{12r}{10} = \frac{6r}{5} \]

### Step 4: Calculate the area of the old and new circles.
The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
The area of the old circle is:
\[ A_{\text{old}} = \pi r^2 \]
The area of the new circle, using the new radius \( R \), is:
\[ A_{\text{new}} = \pi R^2 = \pi \left(\frac{6r}{5}\right)^2 = \pi \left(\frac{36r^2}{25}\right) = \frac{36\pi r^2}{25} \]

### Step 5: Calculate the increase in area.
The increase in area is given by:
\[ \text{Increase} = A_{\text{new}} - A_{\text{old}} = \frac{36\pi r^2}{25} - \pi r^2 \]
To combine these, express \( \pi r^2 \) with a common denominator:
\[ \pi r^2 = \frac{25\pi r^2}{25} \]
Thus,
\[ \text{Increase} = \frac{36\pi r^2}{25} - \frac{25\pi r^2}{25} = \frac{(36 - 25)\pi r^2}{25} = \frac{11\pi r^2}{25} \]

### Step 6: Calculate the percentage increase in area.
The percentage increase in area is given by:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{A_{\text{old}}}\right) \times 100 = \left(\frac{\frac{11\pi r^2}{25}}{\pi r^2}\right) \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = \left(\frac{11}{25}\right) \times 100 = 44\% \]

### Conclusion:
Thus, the area of the circle increases by **44%** when the circumference is increased by 20%.

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