If the circumference of a circle is 3 cm, then its area, in `cm^(2)` is

Options

A. `(9 pi)/( 4)`
B. `(3 pi)/( 2)`
C. `(9)/( 4 pi)`
D. `(4)/(9 pi)`

Answer

To solve the problem of finding the area of a circle given its circumference, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
We know that the circumference of the circle is \(3 \, \text{cm}\).

2. **Use the Formula for Circumference**:
The formula for the circumference \(C\) of a circle is given by:
\[ C = 2\pi r \]
where \(r\) is the radius of the circle.

3. **Set Up the Equation**:
Since the circumference is \(3 \, \text{cm}\), we can set up the equation:
\[ 2\pi r = 3 \]

4. **Solve for the Radius \(r\)**:
To find the radius, we rearrange the equation:
\[ r = \frac{3}{2\pi} \]

5. **Use the Formula for Area**:
The area \(A\) of a circle is given by the formula:
\[ A = \pi r^2 \]

6. **Substitute the Value of \(r\)**:
Now, we substitute the value of \(r\) into the area formula:
\[ A = \pi \left(\frac{3}{2\pi}\right)^2 \]

7. **Calculate \(r^2\)**:
First, calculate \(r^2\):
\[ r^2 = \left(\frac{3}{2\pi}\right)^2 = \frac{9}{4\pi^2} \]

8. **Substitute \(r^2\) into the Area Formula**:
Now substitute \(r^2\) back into the area formula:
\[ A = \pi \cdot \frac{9}{4\pi^2} \]

9. **Simplify the Expression**:
Simplifying the expression gives:
\[ A = \frac{9}{4\pi} \]

10. **Final Answer**:
Therefore, the area of the circle is:
\[ A = \frac{9}{4\pi} \, \text{cm}^2 \]