The problem asks for the length of the arc $s$ cut off by the central angle $\theta = 30^\circ$ in a circle of radius $r = 4 \text{ mm}$.

Formula to Find the Arc Length:

The formula to calculate the arc length $s$ is: $s = r \theta$ where $r$ is the radius of the circle, and $\theta$ is the central angle in radians.

Step 1: Convert the Angle to Radians

Since the angle is given in degrees, we need to convert it to radians. The conversion formula is: $\theta \text{(radians)} = \theta \text{(degrees)} \times \frac{\pi}{180^\circ}$

So, for $\theta = 30^\circ$: $\theta = 30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6} \, \text{radians}$

Step 2: Calculate the Arc Length

Now, substitute $r = 4 \text{ mm}$ and $\theta = \frac{\pi}{6}$ into the arc length formula: $s = 4 \times \frac{\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \, \text{mm}$

Thus, the arc length is: $s = \frac{2\pi}{3} \approx 2.094 \, \text{mm}$

Would you like more details on this calculation or have any specific questions?

Related Questions:

1. How do you convert a radian measure back into degrees?
2. What is the formula for calculating the area of a sector for a given central angle?
3. How does the arc length change if the radius of the circle is doubled?
4. What happens to the arc length if the angle $\theta$ is halved?
5. How would you calculate the perimeter of a sector formed by the central angle?

Tip:

Always ensure that angles are in radians when using formulas related to arc length and sector area.