To solve the problem, we will use the formula for the length of the arc of a circle:

$\text{Arc length} = r \cdot \theta$

Here:

• $r$ is the radius of the circle (in this case, the length of the pendulum, 12 cm).
• $\theta$ is the central angle subtended by the arc, expressed in radians.

Step 1: Convert the angle from degrees to radians

The relationship between degrees and radians is: $\theta_{\text{radians}} = \theta_{\text{degrees}} \cdot \frac{\pi}{180}$ For $\theta = 135^\circ$: $\theta_{\text{radians}} = 135 \cdot \frac{\pi}{180} = \frac{3\pi}{4} \, \text{radians}.$

Step 2: Calculate the arc length

Substituting $r = 12 \, \text{cm}$ and $\theta = \frac{3\pi}{4}$ radians into the formula: $\text{Arc length} = 12 \cdot \frac{3\pi}{4} = 9\pi \, \text{cm}.$

Using $\pi \approx 3.1416$: $\text{Arc length} \approx 9 \cdot 3.1416 = 28.27 \, \text{cm}.$

Final Answer:

The pendulum's path is approximately 28.27 cm long.

Would you like more clarification on this calculation or related concepts?

Here are 5 related questions to expand your understanding:

1. How does the arc length formula change if the angle is given directly in radians?
2. Can you determine the pendulum's path if the swing angle was $90^\circ$?
3. How would you calculate the area of the sector formed by the pendulum's swing?
4. What happens to the arc length if the pendulum's length is doubled but the angle remains the same?
5. How can we verify the answer using a geometric approach?

Tip: Always check the units of the angle (degrees or radians) before substituting it into the arc length formula