The circumference of a circle is 128 m. What is the area of the circle ?

Options

A. `4096/pi`
B. `5056/pi`
C. `5550/pi`
D. none of these

Answer

To find the area of a circle when the circumference is given, we can follow these steps:

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

### Step 2: Substitute the given circumference into the formula
We know that the circumference is 128 m, so we can set up the equation:
\[ 2\pi r = 128 \]

### Step 3: Solve for the radius (r)
To find the radius, we can rearrange the equation:
\[ r = \frac{128}{2\pi} \]
\[ r = \frac{64}{\pi} \]

### Step 4: Use the radius to find the area
The formula for the area (A) of a circle is given by:
\[ A = \pi r^2 \]
Now, substitute the value of \( r \) we found:
\[ A = \pi \left(\frac{64}{\pi}\right)^2 \]

### Step 5: Simplify the area expression
Calculating \( r^2 \):
\[ r^2 = \left(\frac{64}{\pi}\right)^2 = \frac{64^2}{\pi^2} = \frac{4096}{\pi^2} \]
Now substitute this back into the area formula:
\[ A = \pi \cdot \frac{4096}{\pi^2} \]

### Step 6: Cancel out \( \pi \)
This simplifies to:
\[ A = \frac{4096}{\pi} \]

### Conclusion
Thus, the area of the circle is:
\[ A = \frac{4096}{\pi} \, \text{m}^2 \]