Find the length of the arc for a circle with a radius of 18 inches and a central angle of 300degrees leave your answer in terms of pi

1 answer

To find the length of an arc of a circle, you can use the formula:

\[ L = r \theta \]

where:

  • \( L \) is the length of the arc,
  • \( r \) is the radius of the circle, and
  • \( \theta \) is the central angle in radians.

First, we need to convert the central angle from degrees to radians. The conversion from degrees to radians is given by:

\[ \theta \text{ (in radians)} = \theta \text{ (in degrees)} \times \frac{\pi}{180} \]

So for a central angle of 300 degrees:

\[ \theta = 300 \times \frac{\pi}{180} = \frac{300\pi}{180} = \frac{5\pi}{3} \text{ radians} \]

Now we can substitute \( r = 18 \) inches and \( \theta = \frac{5\pi}{3} \) radians into the arc length formula:

\[ L = 18 \times \frac{5\pi}{3} \]

Calculating this gives:

\[ L = \frac{18 \times 5\pi}{3} \]

Simplifying further:

\[ L = \frac{90\pi}{3} = 30\pi \]

Thus, the length of the arc is

\[ \boxed{30\pi} \text{ inches.} \]