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Find the exact length of the arc made by the central angle [tex]\(\theta = 60^{\circ}\)[/tex] in a circle of radius [tex]\(r = 132 \text{ cm}\)[/tex].

Note: Enter the exact, fully simplified answer.
[tex]\[
s = \square \text{ cm}
\][/tex]


Sagot :

To find the exact length of the arc created by a central angle [tex]\(\theta = 60^\circ\)[/tex] in a circle with a radius of [tex]\(r = 132\)[/tex] cm, we can follow these steps:

1. Convert the central angle from degrees to radians:
A central angle in degrees can be converted to radians by using the formula:
[tex]\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \][/tex]
For [tex]\(\theta = 60^\circ\)[/tex]:
[tex]\[ \theta_{\text{radians}} = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \][/tex]

2. Calculate the arc length:
The formula for the arc length [tex]\(s\)[/tex] of a circle is given by:
[tex]\[ s = r \times \theta_{\text{radians}} \][/tex]
Substituting the known values [tex]\(r = 132\)[/tex] cm and [tex]\(\theta_{\text{radians}} = \frac{\pi}{3}\)[/tex]:
[tex]\[ s = 132 \times \frac{\pi}{3} \][/tex]

3. Simplify the expression:
Simplify the multiplication:
[tex]\[ s = \frac{132 \pi}{3} = 44 \pi \][/tex]

Therefore, the exact length of the arc is:
[tex]\[ s = 44 \pi \text{ cm} \][/tex]