If the radius of a circle is increased by 50%, its area is increased by :

Options

A. `125%`
B. `100%`
C. `75%`
D. `50%`

Answer

To solve the problem of how much the area of a circle increases when the radius is increased by 50%, we can follow these steps:

### Step 1: Understand the initial area of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Calculate the initial area
Let the initial radius be \( r \). Therefore, the initial area \( A_1 \) is:
\[ A_1 = \pi r^2 \]

### Step 3: Determine the new radius after the increase
If the radius is increased by 50%, the new radius \( r' \) can be calculated as:
\[ r' = r + 0.5r = 1.5r \]

### Step 4: Calculate the new area with the increased radius
Now, we can calculate the new area \( A_2 \) using the new radius:
\[ A_2 = \pi (r')^2 = \pi (1.5r)^2 \]
\[ A_2 = \pi (2.25r^2) = 2.25\pi r^2 \]

### Step 5: Find the increase in area
The increase in area \( \Delta A \) can be calculated as:
\[ \Delta A = A_2 - A_1 = 2.25\pi r^2 - \pi r^2 \]
\[ \Delta A = (2.25 - 1)\pi r^2 = 1.25\pi r^2 \]

### Step 6: Calculate the percentage increase in area
To find the percentage increase in area, we can use the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta A}{A_1} \right) \times 100 \]
Substituting the values we found:
\[ \text{Percentage Increase} = \left( \frac{1.25\pi r^2}{\pi r^2} \right) \times 100 = 1.25 \times 100 = 125\% \]

### Final Answer
Thus, if the radius of a circle is increased by 50%, its area is increased by **125%**.

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