The circumference of a circle is [tex]$6 \pi$[/tex] inches. What is the area of the circle?

A. [tex]$3 \pi$[/tex] in[tex][tex]$^2$[/tex][/tex]
B. [tex]$9 \pi$[/tex] in[tex]$^2$[/tex]
C. [tex][tex]$12 \pi$[/tex][/tex] in[tex]$^2$[/tex]
D. [tex]$36 \pi$[/tex] in[tex][tex]$^2$[/tex][/tex]



Answer :

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].