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The problem in the image involves a geometric shape where six equilateral triangles form a star pattern around a regular hexagon in the center. Here's the breakdown:

- The perimeter of the central hexagon is 24.
- The side length of the hexagon is half the side length of each equilateral triangle.

### Step-by-Step Solution:

1. **Hexagon Side Length:**
   The perimeter of the hexagon is given as 24. Since a regular hexagon has six equal sides, we divide the perimeter by 6 to find the side length of the hexagon.
   \[
   \text{Side length of hexagon} = \frac{24}{6} = 4
   \]

2. **Triangle Side Length:**
   The problem states that the side length of the hexagon is half the side length of each equilateral triangle. Therefore, the side length of each equilateral triangle is:
   \[
   \text{Side length of triangle} = 2 \times 4 = 8
   \]

3. **Perimeter of the Exterior:**
   The star pattern consists of six equilateral triangles. To find the perimeter of the entire star shape, we must consider the outer edges of these six triangles.

   Each triangle has three sides, but two sides of each triangle are shared with adjacent triangles (forming the inner boundary of the star). Therefore, only the outer side of each triangle contributes to the perimeter.

   Since there are six triangles, and each triangle contributes one side to the outer perimeter, the total perimeter is:
   \[
   \text{Perimeter of the exterior} = 6 \times \text{Side length of triangle} = 6 \times 8 = 48
   \]

### Final Answer:
The perimeter of the exterior of the diagram is **48**.

Would you like more details or have any further questions? Here are some related problems you might be interested in:

1. How do you calculate the area of an equilateral triangle?
2. What is the formula for the perimeter of a regular polygon?
3. Can you find the area of the star-shaped figure?
4. How would the perimeter change if the hexagon’s perimeter was different?
5. How can the properties of symmetry help in solving geometric problems like this one?

**Tip:** When solving geometry problems, visualizing and breaking complex shapes into simpler ones (like triangles or regular polygons) can make the problem more manageable.

In the diagram below, six equilateral triangles are arranged in a ring such that a regular hexagon is formed in the middle. Each side length of the hexagon is half the side length of each triangle. Given that the perimeter of the hexagon is 24, what is the perimeter of the exterior of the diagram?

Learn how to calculate the perimeter of a star-shaped diagram formed by six equilateral triangles surrounding a regular hexagon. This problem involves understanding the properties of regular polygons and applying perimeter formulas. Suitable for students in Grades 7-9.

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