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Let's break down this problem step-by-step to find the length of the major arc \( DFE \).

### Given:
- Radius of the circle \( r = 4 \, \text{m} \).
- \( m \angle DCE = 130^\circ \), which is the central angle for the minor arc \( DE \).
- We are tasked with finding the length of the **major arc** \( DFE \).

### Steps:
1. **Determine the central angle for the major arc \( DFE \):**
   The total angle in a circle is \( 360^\circ \). The central angle for the major arc is:
   \[
   360^\circ - 130^\circ = 230^\circ
   \]

2. **Use the formula for the arc length:**
   The length of an arc is given by:
   \[
   \text{Arc length} = \left(\frac{\theta}{360^\circ}\right) \times 2\pi r
   \]
   where \( \theta \) is the central angle (in degrees) and \( r \) is the radius of the circle.

3. **Substitute the known values:**
   \[
   \text{Arc length of major arc } DFE = \left(\frac{230^\circ}{360^\circ}\right) \times 2\pi (4 \, \text{m})
   \]

4. **Simplify the expression:**
   \[
   \text{Arc length of major arc } DFE = \frac{230}{360} \times 8\pi \, \text{m}
   \]
   Simplify the fraction \( \frac{230}{360} \):
   \[
   \frac{230}{360} = \frac{23}{36}
   \]
   Now, substitute it back:
   \[
   \text{Arc length of major arc } DFE = \frac{23}{36} \times 8\pi \, \text{m}
   \]

5. **Calculate the final answer:**
   \[
   \text{Arc length of major arc } DFE = \frac{184}{36} \pi \, \text{m}
   \]
   Simplify the fraction:
   \[
   \frac{184}{36} = \frac{92}{18} = \frac{46}{9}
   \]

   So, the length of the major arc \( DFE \) is:
   \[
   \boxed{\frac{46\pi}{9} \, \text{m}}
   \]

Would you like further details on this solution?

### Related Questions:
1. What is the formula for finding the circumference of a circle?
2. How do you calculate the length of a minor arc if the central angle is given?
3. How would the solution change if the radius were doubled?
4. What is the relationship between the central angle and the arc length?
5. How do you convert degrees to radians when calculating arc length?

**Tip:** Always ensure the central angle is correctly identified as either minor or major, as this significantly affects the calculation.

The circle below has center C, and its radius is 4 m. Given that ∠DCE = 130°, find the length of the major arc DFE. Give an exact answer in terms of π, and be sure to include the correct unit in your answer.

Learn how to calculate the length of the major arc DFE in a circle with a radius of 4 meters and a central angle of 130°. This problem involves using the arc length formula (θ/360) × 2πr to find the exact solution in terms of π. Suitable for students in Grades 9-12.

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