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To calculate the length of the shape \( ABCDEA \) in the given figure, we need to focus on the different components that form the perimeter.

### Components of the length:
1. **Arc AE**: This is part of a circle with a central angle of \( 150^\circ \).
2. **Line segments**: 
   - \( BC \)
   - \( CD \) (Given as \( 14 \, \text{m} \))
   - \( DE \)

#### Steps for calculation:
1. **Arc length \( AE \)**:
   - The arc length of a circle is given by the formula:
     \[
     L = \theta \times r
     \]
     where:
     - \( \theta \) is the central angle in radians,
     - \( r \) is the radius of the circle.

     Since \( \theta = 150^\circ \), we convert this to radians:
     \[
     \theta = \frac{150 \times \pi}{180} = \frac{5\pi}{6} \, \text{radians}.
     \]
     The radius of the arc (from the diagram, the distance from \( A \) to \( E \)) needs to be determined from the geometry of the figure.

2. **Straight line lengths**: 
   - \( BC \) is part of the structure. We need to estimate or calculate it based on any possible geometric relations in the diagram. Since it might not be directly given, we'll need to interpret this based on the proportions of other segments.

Let me first extract more details from the diagram and run any necessary calculations for the arc length and line segments.

I'll need to estimate certain missing values or relationships based on the figure, such as the radius for the arc AE.

Using an assumed radius of \( 7 \, \text{m} \) for the arc \( AE \):

- The arc length \( AE \) is approximately \( 18.33 \, \text{m} \).
- The straight line segment \( CD \) is given as \( 14 \, \text{m} \).

Now, we need to consider the lengths of other segments like \( BC \) and \( DE \), which might involve geometric relations or estimations from the diagram. If these lengths are also based on similar radius or structural symmetry, they can be estimated to complete the perimeter.

Would you like to make any assumptions or provide additional details for these line segments?

Here are some follow-up steps you might consider:
1. Do you have any hints for the line segment \( BC \)?
2. Would you like to adjust the radius \( r \) based on your knowledge of the figure?
3. Shall we calculate \( DE \) similarly if the radius applies there?
4. Should we look for symmetry to simplify the perimeter calculation?
5. Do you want the total perimeter with the assumptions made?

**Tip:** Always double-check your assumptions for unknown values in geometric problems.

Find the length of the shape ABCDEA in the given figure.

Learn how to calculate the perimeter of the shape ABCDEA, which includes an arc with a 150-degree angle and straight line segments, using geometry formulas such as the arc length formula and perimeter addition. Suitable for students in Grades 9-12.

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