If the circumference of a circle increases from `4pi` to `8pi` what change occurs in its area? 

Options

A. It doubles
B. It triples
C. It quadruples
D. It is halved

Answer

To solve the problem of how the area of a circle changes when its circumference increases from \(4\pi\) to \(8\pi\), we can follow these steps:

### Step 1: Find the radius when the circumference is \(4\pi\).
The formula for the circumference \(C\) of a circle is given by:
\[ C = 2\pi r \]
Setting \(C = 4\pi\):
\[ 4\pi = 2\pi r \]
Dividing both sides by \(2\pi\):
\[ r = \frac{4\pi}{2\pi} = 2 \]

### Step 2: Calculate the area when the radius is \(2\).
The area \(A\) of a circle is given by:
\[ A = \pi r^2 \]
Substituting \(r = 2\):
\[ A = \pi (2^2) = \pi \cdot 4 = 4\pi \]

### Step 3: Find the radius when the circumference is \(8\pi\).
Using the same formula for circumference:
\[ C = 2\pi r \]
Setting \(C = 8\pi\):
\[ 8\pi = 2\pi r \]
Dividing both sides by \(2\pi\):
\[ r = \frac{8\pi}{2\pi} = 4 \]

### Step 4: Calculate the area when the radius is \(4\).
Using the area formula again:
\[ A = \pi r^2 \]
Substituting \(r = 4\):
\[ A = \pi (4^2) = \pi \cdot 16 = 16\pi \]

### Step 5: Determine the change in area.
The initial area when the circumference was \(4\pi\) was \(4\pi\), and the new area when the circumference is \(8\pi\) is \(16\pi\). The change in area is:
\[ \text{Change in area} = 16\pi - 4\pi = 12\pi \]

### Conclusion
The area increases from \(4\pi\) to \(16\pi\), which is a quadrupling of the area.