To calculate the area of the circle given its circumference, we can use the formulas for circumference and area. The formulas are:
- Circumference \( C = 2\pi r \)
- Area \( A = \pi r^2 \)
First, we need to find the radius \( r \) from the given circumference \( C = 81.64 \) inches.
Using the circumference formula:
\[ C = 2\pi r \]
Substituting the value of \( C \):
\[ 81.64 = 2 \times 3.14 \times r \]
Now, we can solve for \( r \):
\[ 81.64 = 6.28r \]
\[ r = \frac{81.64}{6.28} \approx 13.0 \text{ inches} \]
Now that we have the radius, we can find the area using the area formula:
\[ A = \pi r^2 \]
Substituting \( r \) and using \( \pi \approx 3.14 \):
\[ A = 3.14 \times (13.0)^2 \]
Calculating \( (13.0)^2 \):
\[ (13.0)^2 = 169.0 \]
Now, plug that back into the area formula:
\[ A = 3.14 \times 169.0 \approx 530.46 \]
Finally, rounding to the nearest tenth:
\[ A \approx 530.5 \text{ square inches} \]
So, the area of the circle is approximately 530.5 square inches.