Pack the circles with diameter equal to 24 inches using square and hexagon.

Packing circles within different geometric patterns like squares and hexagons leads to varying levels of space efficiency. Let's analyze how the packing works for both square and hexagonal arrangements, using circles of diameter 24 inches.

### 1. Square Packing

In a **square packing** arrangement, circles are arranged in rows and columns. Each circle is positioned directly next to its neighboring circles, both horizontally and vertically. 

- The distance between the centers of any two adjacent circles (both horizontally and vertically) is equal to the diameter of the circles.
- Therefore, for circles with a diameter of 24 inches, the center-to-center distance is also 24 inches.
  
This type of arrangement is relatively inefficient in terms of space usage because there are empty spaces (gaps) between the circles, especially along the diagonal lines between them.

**Efficiency of Square Packing**:
- The area of the square containing one circle (with side length 24 inches) is:
  \[
  A_{\text{square}} = 24 \times 24 = 576 \, \text{square inches}
  \]
- The area of one circle is:
  \[
  A_{\text{circle}} = \pi \times \left( \frac{24}{2} \right)^2 = \pi \times 12^2 = 144\pi \, \text{square inches} \approx 452.39 \, \text{square inches}
  \]
- The packing efficiency (the percentage of the area covered by circles) is:
  \[
  \text{Efficiency}_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} \times 100 = \frac{144\pi}{576} \times 100 \approx 78.54\%
  \]

### 2. Hexagonal Packing

In a **hexagonal packing** arrangement, circles are packed in a staggered or honeycomb-like structure. This type of packing is more efficient than square packing because the circles are closer together, reducing the gaps between them.

- In hexagonal packing, each circle touches six others (instead of four in the square arrangement), and the distance between the centers of two adjacent circles is still equal to the diameter (24 inches).
  
**Efficiency of Hexagonal Packing**:
- The area occupied by one circle and its surrounding hexagonal arrangement can be calculated by looking at the geometry of the hexagon.
- The hexagonal packing density is a known constant and is approximately 90.69%.
  
This means that about **90.69%** of the available area is filled by circles in a hexagonal arrangement.

### Conclusion
- **Square Packing Efficiency**: ~78.54%
- **Hexagonal Packing Efficiency**: ~90.69%

Thus, hexagonal packing is more efficient than square packing when arranging circles with a diameter of 24 inches. You can fit more circles in a given space using the hexagonal arrangement.

Would you like more details or clarifications on this?

### Related Questions:
1. How do the formulas for packing efficiency differ between square and hexagonal packing?
2. What is the maximum number of circles that can fit inside a given rectangular area using square packing?
3. How can the packing density be improved further for irregular shapes?
4. Can the packing efficiency change if the diameter of the circles changes?
5. How does 3D sphere packing differ from 2D circle packing?

### Tip:
Hexagonal packing is commonly used in nature (e.g., honeycombs) because it maximizes space efficiency.

Discover the difference between square and hexagonal packing arrangements for circles with a 24-inch diameter. Analyze the space efficiency using geometry, including key formulas for area and packing density. Suitable for Grades 9-12 students interested in understanding space optimization.

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