Question

If a circle has a radius of 6 cm and a central angle of 5π4
5
𝜋
4
, then what is the length of the arc created by the angle?

Answer 1

To find the length of the arc corresponding to a central angle in a circle, you can use the formula:

\[
L = r \theta
\]

where:
- \( L \) is the arc length,
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle in radians.

For this particular problem:
- The radius \( r = 6 \) cm,
- The central angle \( \theta = \frac{5\pi}{4} \) radians.

Now, substituting the values into the formula:

\[
L = 6 \cdot \frac{5\pi}{4}
\]

Calculating this step by step:

1. Multiply the radius by the angle:
\[
L = \frac{30\pi}{4}
\]

2. Simplify:
\[
L = \frac{15\pi}{2} \text{ cm}
\]

Thus, the length of the arc is:

\[
\boxed{\frac{15\pi}{2}} \text{ cm}
\]