Question
If a circle has a central angle of 2π3 and a radius of 27 millimeters, how long is the arc formed by that angle?(1 point) Responses 5π18 mm Start Fraction 5 pi over 18 End Fraction mm 67.5π2 mm Start Fraction 67.5 pi over 2 End Fraction mm 15π mm 15 pi mm 18π mm
Answers
Answered by
GPT-4o mini
To find the length of an arc formed by a central angle in a circle, you can use the formula:
\[
\text{Arc length} = r \cdot \theta
\]
where \( r \) is the radius and \( \theta \) is the central angle in radians.
In this case:
- The radius \( r = 27 \) mm
- The central angle \( \theta = \frac{2\pi}{3} \) radians
Now, substituting these values into the formula:
\[
\text{Arc length} = 27 \cdot \frac{2\pi}{3}
\]
Calculating this:
\[
\text{Arc length} = 27 \cdot \frac{2\pi}{3} = \frac{54\pi}{3} = 18\pi
\]
Therefore, the length of the arc is:
\[
\boxed{18\pi \text{ mm}}
\]
\[
\text{Arc length} = r \cdot \theta
\]
where \( r \) is the radius and \( \theta \) is the central angle in radians.
In this case:
- The radius \( r = 27 \) mm
- The central angle \( \theta = \frac{2\pi}{3} \) radians
Now, substituting these values into the formula:
\[
\text{Arc length} = 27 \cdot \frac{2\pi}{3}
\]
Calculating this:
\[
\text{Arc length} = 27 \cdot \frac{2\pi}{3} = \frac{54\pi}{3} = 18\pi
\]
Therefore, the length of the arc is:
\[
\boxed{18\pi \text{ mm}}
\]