Answer :
To determine the area of the circle given its circumference, we can follow these steps:
### Step 1: Recall the formula for the circumference of a circle
The circumference \( C \) of a circle is given by:
[tex]\[ C = 2 \pi r \][/tex]
where \( r \) is the radius of the circle.
### Step 2: Plug in the given circumference and solve for the radius
We are given that the circumference \( C \) is \( 3 \pi \). Therefore:
[tex]\[ 3 \pi = 2 \pi r \][/tex]
To solve for \( r \), we divide both sides of the equation by \( 2 \pi \):
[tex]\[ r = \frac{3 \pi}{2 \pi} = \frac{3}{2} = 1.5 \text{ inches} \][/tex]
### Step 3: Recall the formula for the area of a circle
The area \( A \) of a circle is given by:
[tex]\[ A = \pi r^2 \][/tex]
### Step 4: Substitute the radius into the area formula
Using the radius \( r = 1.5 \) derived in Step 2:
[tex]\[ A = \pi (1.5)^2 \][/tex]
### Step 5: Calculate the area
Firstly, calculate the square of the radius:
[tex]\[ (1.5)^2 = 1.5 \times 1.5 = 2.25 \][/tex]
Now multiply by \( \pi \):
[tex]\[ A = \pi \times 2.25 = 2.25 \pi \][/tex]
### Conclusion
The area of the circle is \( 2.25 \pi \) square inches.
Hence, the correct answer is:
[tex]\[ \boxed{2.25 \pi \text{ in}^2} \][/tex]
### Step 1: Recall the formula for the circumference of a circle
The circumference \( C \) of a circle is given by:
[tex]\[ C = 2 \pi r \][/tex]
where \( r \) is the radius of the circle.
### Step 2: Plug in the given circumference and solve for the radius
We are given that the circumference \( C \) is \( 3 \pi \). Therefore:
[tex]\[ 3 \pi = 2 \pi r \][/tex]
To solve for \( r \), we divide both sides of the equation by \( 2 \pi \):
[tex]\[ r = \frac{3 \pi}{2 \pi} = \frac{3}{2} = 1.5 \text{ inches} \][/tex]
### Step 3: Recall the formula for the area of a circle
The area \( A \) of a circle is given by:
[tex]\[ A = \pi r^2 \][/tex]
### Step 4: Substitute the radius into the area formula
Using the radius \( r = 1.5 \) derived in Step 2:
[tex]\[ A = \pi (1.5)^2 \][/tex]
### Step 5: Calculate the area
Firstly, calculate the square of the radius:
[tex]\[ (1.5)^2 = 1.5 \times 1.5 = 2.25 \][/tex]
Now multiply by \( \pi \):
[tex]\[ A = \pi \times 2.25 = 2.25 \pi \][/tex]
### Conclusion
The area of the circle is \( 2.25 \pi \) square inches.
Hence, the correct answer is:
[tex]\[ \boxed{2.25 \pi \text{ in}^2} \][/tex]
