Question
If a circle has a circumference of 20π and a central angle of 90 degrees is drawn, what is the length of the minor arc?
Answers
GPT-4o mini
To find the length of the minor arc corresponding to a central angle of 90 degrees in a circle with a circumference of \(20\pi\), we can use the formula for the length of an arc:
\[
\text{Arc Length} = \left(\frac{\theta}{360^\circ}\right) \times C
\]
where:
- \(\theta\) is the central angle in degrees,
- \(C\) is the circumference of the circle.
Given:
- \(C = 20\pi\)
- \(\theta = 90^\circ\)
We can substitute these values into the formula:
\[
\text{Arc Length} = \left(\frac{90^\circ}{360^\circ}\right) \times 20\pi
\]
Calculating the fraction:
\[
\frac{90}{360} = \frac{1}{4}
\]
Now substituting this back into the formula:
\[
\text{Arc Length} = \frac{1}{4} \times 20\pi = 5\pi
\]
Thus, the length of the minor arc is
\[
\boxed{5\pi}
\]
\[
\text{Arc Length} = \left(\frac{\theta}{360^\circ}\right) \times C
\]
where:
- \(\theta\) is the central angle in degrees,
- \(C\) is the circumference of the circle.
Given:
- \(C = 20\pi\)
- \(\theta = 90^\circ\)
We can substitute these values into the formula:
\[
\text{Arc Length} = \left(\frac{90^\circ}{360^\circ}\right) \times 20\pi
\]
Calculating the fraction:
\[
\frac{90}{360} = \frac{1}{4}
\]
Now substituting this back into the formula:
\[
\text{Arc Length} = \frac{1}{4} \times 20\pi = 5\pi
\]
Thus, the length of the minor arc is
\[
\boxed{5\pi}
\]