To find the length of the arc intercepted by a central angle $\theta$ in a circle of radius $r$, we use the formula:

$\text{Arc Length} = r \times \theta$

Given:

• $r = 12.5$ cm
• $\theta = \frac{2\pi}{5}$ radians

Let's substitute these values into the formula:

$\text{Arc Length} = 12.5 \times \frac{2\pi}{5}$

Simplifying:

$\text{Arc Length} = 12.5 \times \frac{2 \times 3.1416}{5}$ $\text{Arc Length} = 12.5 \times \frac{6.2832}{5}$ $\text{Arc Length} = 12.5 \times 1.25664$ $\text{Arc Length} = 15.708 \text{ cm}$

Rounding to three significant digits:

$\text{Arc Length} \approx 15.7 \text{ cm}$

So, the length of the arc is approximately 15.7 cm.

Do you want more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the arc length change if the radius is doubled?
2. What is the formula for the area of a sector given the radius and angle $\theta$?
3. How does the arc length change if $\theta$ is expressed in degrees instead of radians?
4. What is the relationship between the arc length and the circumference of the circle?
5. How can you find the length of the chord that subtends the same angle $\theta$ at the center?

Tip: When working with angles in radians, remember that $2\pi$ radians is equivalent to a full circle (360 degrees).