Question
The radius of a circle is 9 feet. What is the length of a 90° arc?
Answers
GPT-4o mini
To find the length of a 90° arc in a circle with a radius of 9 feet, we can use the formula for the length of an arc, which is given by:
\[
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
\]
where \(\theta\) is the angle in degrees and \(r\) is the radius of the circle.
In this case, \(\theta = 90^\circ\) and \(r = 9\) feet. Plugging in these values:
\[
\text{Arc Length} = \frac{90}{360} \times 2\pi \times 9
\]
First, simplify \(\frac{90}{360}\):
\[
\frac{90}{360} = \frac{1}{4}
\]
Now, substituting this back into the formula:
\[
\text{Arc Length} = \frac{1}{4} \times 2\pi \times 9
\]
Calculating this gives:
\[
\text{Arc Length} = \frac{1}{4} \times 18\pi = \frac{18\pi}{4} = \frac{9\pi}{2}
\]
To get an approximate decimal value, we can use \(\pi \approx 3.14\):
\[
\text{Arc Length} \approx \frac{9 \times 3.14}{2} \approx \frac{28.26}{2} \approx 14.13
\]
Thus, the length of the 90° arc is:
\[
\boxed{\frac{9\pi}{2}} \text{ feet} \quad \text{(approximately 14.13 feet)}
\]
\[
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
\]
where \(\theta\) is the angle in degrees and \(r\) is the radius of the circle.
In this case, \(\theta = 90^\circ\) and \(r = 9\) feet. Plugging in these values:
\[
\text{Arc Length} = \frac{90}{360} \times 2\pi \times 9
\]
First, simplify \(\frac{90}{360}\):
\[
\frac{90}{360} = \frac{1}{4}
\]
Now, substituting this back into the formula:
\[
\text{Arc Length} = \frac{1}{4} \times 2\pi \times 9
\]
Calculating this gives:
\[
\text{Arc Length} = \frac{1}{4} \times 18\pi = \frac{18\pi}{4} = \frac{9\pi}{2}
\]
To get an approximate decimal value, we can use \(\pi \approx 3.14\):
\[
\text{Arc Length} \approx \frac{9 \times 3.14}{2} \approx \frac{28.26}{2} \approx 14.13
\]
Thus, the length of the 90° arc is:
\[
\boxed{\frac{9\pi}{2}} \text{ feet} \quad \text{(approximately 14.13 feet)}
\]