Question
If a circle has a circumference of 20π and a central angle of 90 degrees is drawn, then 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. The length of an arc \( L \) can be calculated using the formula:
\[
L = \frac{\theta}{360^\circ} \times C
\]
where:
- \( \theta \) is the central angle in degrees,
- \( C \) is the circumference of the circle.
In this case:
- \( \theta = 90^\circ \)
- \( C = 20\pi \)
Now plug in the values:
\[
L = \frac{90}{360} \times 20\pi
\]
First, simplify \( \frac{90}{360} \):
\[
\frac{90}{360} = \frac{1}{4}
\]
Now substitute this back into the equation:
\[
L = \frac{1}{4} \times 20\pi = 5\pi
\]
Thus, the length of the minor arc is
\[
\boxed{5\pi}
\]
\[
L = \frac{\theta}{360^\circ} \times C
\]
where:
- \( \theta \) is the central angle in degrees,
- \( C \) is the circumference of the circle.
In this case:
- \( \theta = 90^\circ \)
- \( C = 20\pi \)
Now plug in the values:
\[
L = \frac{90}{360} \times 20\pi
\]
First, simplify \( \frac{90}{360} \):
\[
\frac{90}{360} = \frac{1}{4}
\]
Now substitute this back into the equation:
\[
L = \frac{1}{4} \times 20\pi = 5\pi
\]
Thus, the length of the minor arc is
\[
\boxed{5\pi}
\]