Question

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

Answer 1

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

\[
\text{Arc Length} = r \cdot \theta
\]

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

In this case, the radius \( r \) is 6 cm, and the central angle \( \theta \) is \( \frac{5\pi}{4} \) radians.

Now, substitute these values into the formula:

\[
\text{Arc Length} = 6 \, \text{cm} \cdot \frac{5\pi}{4}
\]

Now calculate the arc length:

\[
\text{Arc Length} = 6 \cdot \frac{5\pi}{4} = \frac{30\pi}{4} = \frac{15\pi}{2} \, \text{cm}
\]

Thus, the length of the arc is:

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

If you need a numerical approximation, you can use \( \pi \approx 3.14 \):

\[
\text{Arc Length} \approx \frac{15 \times 3.14}{2} \approx 23.55 \, \text{cm}
\]

So, the length of the arc is \( \frac{15\pi}{2} \) cm or approximately 23.55 cm.