IF the radius of a circle is doubled, by how much will the area ?

Options

A. 2 times
B. 3 times
C. 4 times
D. 5 times

Answer

To solve the question "If the radius of a circle is doubled, by how much will the area increase?", we can follow these steps:

### Step-by-Step Solution:

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

**Hint**: Remember the formula for the area of a circle, which involves the radius squared.

2. **Calculate the area with the original radius**:
Let the original radius be \( r \). The area of the circle with this radius is:
\[ A_1 = \pi r^2 \]

**Hint**: Substitute the original radius into the area formula to find the initial area.

3. **Determine the new radius when it is doubled**:
If the radius is doubled, the new radius becomes:
\[ r' = 2r \]

**Hint**: Think about how doubling the radius affects its value.

4. **Calculate the area with the new radius**:
Now, we can find the area of the circle with the new radius:
\[ A_2 = \pi (2r)^2 \]
Simplifying this gives:
\[ A_2 = \pi (4r^2) = 4\pi r^2 \]

**Hint**: Remember to square the entire term when substituting the new radius into the area formula.

5. **Compare the two areas**:
Now we compare the original area \( A_1 \) and the new area \( A_2 \):
- Original area: \( A_1 = \pi r^2 \)
- New area: \( A_2 = 4\pi r^2 \)

**Hint**: Look at how the new area relates to the original area by factoring.

6. **Calculate the increase in area**:
The increase in area can be calculated as:
\[ \text{Increase in Area} = A_2 - A_1 = 4\pi r^2 - \pi r^2 = (4 - 1)\pi r^2 = 3\pi r^2 \]

**Hint**: Subtract the original area from the new area to find the increase.

7. **Determine the factor of increase**:
To find out by how much the area has increased in terms of the original area, we can express the new area in relation to the original area:
\[ \frac{A_2}{A_1} = \frac{4\pi r^2}{\pi r^2} = 4 \]

**Hint**: Dividing the new area by the original area shows how many times larger the new area is.

### Final Answer:
The area of the circle increases by a factor of 4 when the radius is doubled.