To find the area of a circle given its circumference, follow these detailed steps:
1. Recall the formula for the circumference of a circle: The circumference [tex]\( C \)[/tex] of a circle is given by [tex]\( C = 2 \pi r \)[/tex], where [tex]\( r \)[/tex] is the radius.
2. Set up the equation with the given circumference: We know that the circumference of the circle is [tex]\( 6 \pi \)[/tex] inches. Set this equal to the circumference formula:
[tex]\[
6 \pi = 2 \pi r
\][/tex]
3. Solve for the radius [tex]\( r \)[/tex]:
[tex]\[
r = \frac{6 \pi}{2 \pi} = 3
\][/tex]
So, the radius of the circle is 3 inches.
4. Recall the formula for the area of a circle: The area [tex]\( A \)[/tex] of a circle is given by [tex]\( A = \pi r^2 \)[/tex].
5. Substitute the radius into the area formula:
[tex]\[
A = \pi (3)^2 = \pi \times 9 = 9 \pi
\][/tex]
6. Conclusion: The area of the circle is [tex]\( 9 \pi \)[/tex] square inches.
Thus, the correct answer is [tex]\( 9 \pi \)[/tex] in. [tex]\( ^2 \)[/tex].
