https://file.aiquickdraw.com/mathbot/file/1725982918627vy60ut85.jpg

To solve for the missing dimension of the regular hexagon, let's analyze the given information:

1. **Hexagon Geometry**: A regular hexagon can be divided into six equilateral triangles. The key dimension provided in the image is the length of the height of one of these triangles, which is given as 64 inches.

2. **Relationship Between Height and Side Length**:
   - In an equilateral triangle, the height \( h \) relates to the side length \( s \) by the formula:
     \[
     h = \frac{\sqrt{3}}{2} s
     \]
   - Given \( h = 64 \) inches, we can solve for \( s \).

3. **Solving for the Side Length \( s \)**:
   \[
   s = \frac{2h}{\sqrt{3}} = \frac{2 \times 64}{\sqrt{3}} \approx \frac{128}{1.732} \approx 73.9 \text{ inches}
   \]

So, the missing dimension, which is the side length \( s \) of the regular hexagon, is approximately **73.9 inches**.

Would you like further details or have any questions?

Here are some related questions you might consider:

1. How is the height of an equilateral triangle derived?
2. What are the other dimensions in a regular hexagon related to the side length?
3. How can you find the perimeter of the hexagon with the given side length?
4. How does the side length relate to the radius of the circumscribed circle?
5. What are the properties of a regular hexagon that make these calculations possible?

**Tip:** Always verify which dimension you're solving for—height, side length, or diagonal—as they relate differently to the geometry of a regular hexagon.

Find the missing dimension of the regular hexagon shown to the right. The height of the hexagon is 64 inches.

Learn how to calculate the missing side length of a regular hexagon using its height of 64 inches. This problem explores key geometric concepts, such as equilateral triangles and hexagon properties. Suitable for high school students in grades 9-12.

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