Question
The circumference of a circle is 19π ft. What is the area, in square feet?
Answers
GPT-4o mini
To find the area of the circle, we first need to determine the radius from the given circumference. The formula for the circumference \( C \) of a circle is:
\[
C = 2\pi r
\]
where \( r \) is the radius. Given that the circumference is \( 19\pi \) ft, we can set up the equation:
\[
2\pi r = 19\pi
\]
To solve for \( r \), we divide both sides by \( 2\pi \):
\[
r = \frac{19\pi}{2\pi} = \frac{19}{2} \text{ ft}
\]
Next, we will calculate the area \( A \) of the circle using the area formula:
\[
A = \pi r^2
\]
Substituting in the value of \( r \):
\[
A = \pi \left( \frac{19}{2} \right)^2
\]
Calculating \( \left( \frac{19}{2} \right)^2 \):
\[
\left( \frac{19}{2} \right)^2 = \frac{361}{4}
\]
Now substituting this back into the area formula:
\[
A = \pi \cdot \frac{361}{4} = \frac{361\pi}{4} \text{ square feet}
\]
Thus, the area of the circle is
\[
\boxed{\frac{361\pi}{4}} \text{ square feet.}
\]
\[
C = 2\pi r
\]
where \( r \) is the radius. Given that the circumference is \( 19\pi \) ft, we can set up the equation:
\[
2\pi r = 19\pi
\]
To solve for \( r \), we divide both sides by \( 2\pi \):
\[
r = \frac{19\pi}{2\pi} = \frac{19}{2} \text{ ft}
\]
Next, we will calculate the area \( A \) of the circle using the area formula:
\[
A = \pi r^2
\]
Substituting in the value of \( r \):
\[
A = \pi \left( \frac{19}{2} \right)^2
\]
Calculating \( \left( \frac{19}{2} \right)^2 \):
\[
\left( \frac{19}{2} \right)^2 = \frac{361}{4}
\]
Now substituting this back into the area formula:
\[
A = \pi \cdot \frac{361}{4} = \frac{361\pi}{4} \text{ square feet}
\]
Thus, the area of the circle is
\[
\boxed{\frac{361\pi}{4}} \text{ square feet.}
\]