To find the area of a circle when given its circumference, we need to follow a series of steps involving some key formulas from geometry.
1. Understand the relationship between circumference and radius:
The formula for the circumference [tex]\( C \)[/tex] of a circle is:
[tex]\[
C = 2\pi r
\][/tex]
Here, [tex]\( C \)[/tex] is the circumference and [tex]\( r \)[/tex] is the radius of the circle. We can rearrange this formula to solve for the radius [tex]\( r \)[/tex]:
[tex]\[
r = \frac{C}{2\pi}
\][/tex]
2. Calculate the radius:
Given:
[tex]\[
C = 56.52 \, \text{inches}
\][/tex]
Using the rearranged formula:
[tex]\[
r = \frac{56.52}{2\pi}
\][/tex]
We know [tex]\( \pi \approx 3.14159 \)[/tex], so:
[tex]\[
r = \frac{56.52}{2 \times 3.14159} \approx \frac{56.52}{6.28318} \approx 9 \, \text{inches}
\][/tex]
3. Find the area using the radius:
The formula for the area [tex]\( A \)[/tex] of a circle is:
[tex]\[
A = \pi r^2
\][/tex]
Now that we have the radius [tex]\( r \approx 9 \)[/tex] inches, we can substitute it into the area formula:
[tex]\[
A = \pi \times 9^2
\][/tex]
Simplifying further:
[tex]\[
A = 3.14159 \times 81 \approx 254.469 \, \text{square inches}
\][/tex]
So, the area of the circle is approximately [tex]\( 254.469 \, \text{square inches} \)[/tex].
